Learning Objectives

*In this chapter, you will learn to:*

- Graph a linear equation.
- Find the slope of a line.
- Determine an equation of a line.
- Solve linear systems.
- Do application problems using linear equations.

Thumbnail: The red and blue lines on this graph have the same slope (gradient); the red and green lines have the same y-intercept (cross the y-axis at the same place). (CC BY-SA 1.0; ElectroKid via Wikipedia)

## Analyzing Linear Equations

Knowing how to write linear equations is an important steping stone on the road to becoming a master mathematician! In this tutorial, you'll practice using a slope and one point to write the equation of the line in standard form.

#### How Do You Find the Slope of a Line from a Graph?

Trying to find the slope of a graphed line? First, identify two points on the line. Then, you could use these points to figure out the slope. In this tutorial, you'll see how to use two points on the line to find the change in 'y' and the change in 'x'. Then, you'll see how to take these values and calculate the slope. Check it out!

#### How Do You Find the X- and Y-Intercepts of a Line in Slope-Intercept Form?

To find the x-intercept of a given linear equation, plug in 0 for 'y' and solve for 'x'. To find the y-intercept, plug 0 in for 'x' and solve for 'y'. In this tutorial, you'll see how to find the x-intercept and the y-intercept for a given linear equation. Check it out!

#### How Do You Write the Equation of a Line in Slope-Intercept Form If You Have Two Points?

Trying to write an equation in slope-intercept form? Have two points on your line? You'll need to find your slope and y-intercept. Watch this tutorial and see what needs to be done to write an equation in slope-intercept form!

#### How Do You Write the Equation of a Line in Slope-Intercept Form from a Word Problem?

Word problems are a great way to see math in the real world! In this tutorial, you'll see how write an equation in slope-intercept form that represents the information given in the word problem. To see how it's done, check out this tutorial!

#### How Do You Write an Equation of a Line in Slope-Intercept Form If You Have One Point and a Parallel Line?

Want to find the slope-intercept form of a line when you're given a point on that line and another line parallel to that line? Remember, parallel lines have the same slope. If you can find the slope of that parallel line, you'll have the slope of your line! In this tutorial, you'll see how to find the slope of your line and use that slope, along with the given point, to write an equation for the line in slope-intercept form. Take a look!

#### How Do You Write an Equation of a Line in Slope-Intercept Form If You Have One Point and a Perpendicular Line?

Want to find the slope-intercept form of a line when you're given a point on that line and another line perpendicular to that line? Remember, perpendicular lines have slopes that are opposite reciprocals of each other. In this tutorial, you'll see how to find the slope using the slope of the perpendicular line. Then, use this slope and the given point to write an equation for the line in slope-intercept form. Check it out!

#### How Do You Put an Equation in Standard Form Into Slope-Intercept or Point-Slope Form?

Looking for some practice converting the equation of a line into different forms? Then this tutorial was made for you! Follow along as this tutorial shows you how to take a linear equation from standard form and convert it into slope-intercept form and point-slope form.

#### How Do You Put an Equation in Slope-Intercept Form Into Standard or Point-Slope Form?

Looking for some practice converting the equation of a line into different forms? Then this tutorial was made for you! Follow along as this tutorial shows you how to take a linear equation from slope-intercept form and convert it into standard form and point-slope form.

#### How Do You Put an Equation in Point-Slope Form Into Standard or Slope-Intercept Form?

Looking for some practice converting the equation of a line into different forms? Then this tutorial was made for you! Follow along as this tutorial shows you how to take a linear equation from point-slope form and convert it into standard form and slope-intercept form.

#### How Do You Find the Slope of a Line from Two Points?

Calculating the slope of a line from two given points? Use the slope formula! This tutorial will show you how!

#### How Do You Find the Y-Intercept of a Line If You Have Another Point and the Slope?

Want some practice finding the y-intercept of a line? In this tutorial, you're given the slope of a line and a point on that line and asked to find the y-intercept. Watch this tutorial and see how the equation for the slope-intercept form of a line is used to figure out the answer!

#### How Do You Find the Constant of Variation from a Direct Variation Equation?

The constant of variation is the number that relates two variables that are directly proportional or inversely proportional to one another. Watch this tutorial to see how to find the constant of variation for a direct variation equation. Take a look!

#### How Do You Write an Equation for Direct Variation Given a Point?

Looking for some practice with direct variation? Watch this tutorial, and get that practice! This tutorial shows you how to take given information and turn it into a direct variation equation. Then, see how to use that equation to find the value of one of the variables.

#### How Do You Write an Equation for Direct Variation from a Table?

Looking for some practice with direct variation? Watch this tutorial, and get that practice! This tutorial shows you how to take a table of values and describe the relation using a direct variation equation.

#### How Do You Find the X-Coordinate of a Point on a Line If You Have Another Point and the Slope?

How do you find the x-coordinate of a point on a line if you have another point and the slope? You'll need to use the slope formula. Watch this tutorial and see how to find this missing coordinate!

#### How Do You Use a Scatter Plot to Find a Positive Correlation?

Got a bunch of data? Trying to figure out if there is a positive, negative, or no correlation? Draw a scatter plot! This tutorial takes you through the steps of creating a scatter plot, drawing a line-of-fit, and determining the correlation, if any. Take a look!

#### How Do You Use a Scatter Plot to Find a Line of Fit?

A line-of-fit is a line that summarizes the trend in a set of data. In this tutorial, you'll see how to graph data on a coordinate plane and draw a line-of-fit for that data. Check it out!

#### How Do You Use a Scatter Plot to Find a Negative Correlation?

Got a bunch of data? Trying to figure out if there is a positive, negative, or no correlation? Draw a scatter plot! This tutorial takes you through the steps of creating a scatter plot, drawing a line-of-fit, and determining the correlation, if any. Take a look!

#### How Do You Use a Scatter Plot to Find No Correlation?

Got a bunch of data? Trying to figure out if there is a positive, negative, or no correlation? Draw a scatter plot! This tutorial takes you through the steps of creating a scatter plot, drawing a line-of-fit, and determining the correlation, if any. Take a look!

#### How Do You Write an Equation for a Vertical Line?

Trying to find the equation of a vertical line that goes through a given point? Remember that vertical lines only have an 'x' value and no 'y' value. Follow along with this tutorial as you see how use the information given to write the equation of a vertical line.

#### How Do You Write an Equation for a Horizontal Line?

Trying to find the equation of a horizontal line that goes through a given point? Remember that vertical lines only have a 'y' value and no 'x' value. Follow along with this tutorial as you see how use the information given to write the equation of a horizontal line.

#### How Do You Graph a Vertical Line?

To graph a vertical line that goes through a given point, first plot that point. Then draw a straight line up and down that goes through the point, and you're done! To see this process in action, watch this tutorial!

#### How Do You Graph a Horizontal Line?

To graph a horizontal line that goes through a given point, first plot that point. Then draw a straight line left and right that goes through the point, and you're done! To see this process in action, watch this tutorial!

#### How Do You Find the Slope of a Line If You Have a Perpendicular Line?

Perpendicular lines have slopes that are opposite reciprocals of each other. To find the slope of a line that is perpendicular to a given equation, find the opposite reciprocal of that slope. Check out this tutorial to learn how!

#### How Do You Find the Slope of a Ramp If You Know the Rise and Run?

Word problems are a great way to see math in action! This tutorial shows you how to solve a word problem involving rise and run by using the slope formula.

#### How Do You Use X- and Y-Intercepts To Graph a Line In Standard Form?

To find the x-intercept of a given linear equation, simply remove the 'y' and solve for 'x'. To find the y-intercept, remove the 'x' and solve for 'y'. In this tutorial, you'll see how to find the x-intercept and the y-intercept for a given linear equation. Check it out!

#### How Do You Graph a Line If You're Given the Slope and the Intercept?

Trying to graph a line from a given slope and y-intercept? Think you need to find an equation first? Think again! In this tutorial, see how to use that given slope and y-intercept to graph the line.

#### How Do You Graph a Line If You're Given the Slope and a Single Point?

Trying to graph a line from a given slope and a point? Think you need to find an equation first? Think again! In this tutorial, see how to use that given slope and point to graph the line.

#### How Do You Write an Equation of a Line in Point-Slope Form If You Have the Slope and One Point?

Trying to write an equation in point-slope form? Got a point on the line and the slope? Plug those values correctly into the point-slope form of a line and you'll have your answer! Watch this tutorial to get all the details!

#### How Do You Write an Equation of a Line in Point-Slope Form If You Have Two Points?

Trying to write an equation in point-slope form? Have two points but no slope? You'll need to use those points to find a slope first. Watch this tutorial and see what needs to be done to write an equation in point-slope form!

#### How Do You Write an Equation of a Line in Point-Slope Form and Standard Form If You Have Two Points?

Get some practice with the point-slope form and standard form of an equation! This tutorial shows you how to use two given points to write an equation in both forms. Take a look!

#### How Do You Write the Equation of a Line in Slope-Intercept Form If You Have the Slope and the Y-Intercept?

Want to write an equation in slope-intercept form? Already have the slope and y-intercept? Perfect! Just correctly plug those values into your equation and you're done! Learn how in this tutorial.

#### What's the Formula for Slope?

When you're dealing with linear equations, you may be asked to find the slope of a line. That's when knowing the slope formula really comes in handy! Learn the formula to find the slope of a line by watching this tutorial.

#### What's the X-Intercept?

When you have a linear equation, the x-intercept is the point where the graph of the line crosses the x-axis. In this tutorial, learn about the x-intercept. Check it out!

#### How Do You Know if Two Lines are Parallel?

Parallel lines are lines that will go on and on forever without ever intersecting. This is because they have the same slope! If you have two linear equations that have the same slope but different y-intercepts, then those lines are parallel to one another!

#### How Do You Know if Two Lines are Perpendicular?

Perpendicular lines intersect at right angles to one another. To figure out if two equations are perpendicular, take a look at their slopes. The slopes of perpendicular lines are opposite reciprocals of each other. Their product is -1! Watch this tutorial and see how to determine if two equations are perpendicular.

#### What Does the Slope of a Line Mean?

You can't learn about linear equations without learning about slope. The slope of a line is the steepness of the line. There are many ways to think about slope. Slope is the rise over the run, the change in 'y' over the change in 'x', or the gradient of a line. Check out this tutorial to learn about slope!

#### What's the Constant of Variation?

The constant of variation is the number that relates two variables that are directly proportional or inversely proportional to one another. But why is it called the constant of variation? This tutorial answers that question, so take a look!

#### What Does Direct Variation Look Like on a Graph?

Want to know what a direct variation looks like graphically? Basically, it's a straight line that goes through the origin. To get a better picture, check out this tutorial!

#### What's a Scatter Plot?

Scatter plots are really useful for graphically showing a bunch of data. By seeing data graphically, you can see patterns or trends in the data. These patterns help researchers to understand how one thing affects another. This can lead to all kinds of breakthroughs! This tutorial gives you a look at the scatter plot. Check it out!

#### What's Positive Correlation?

Looking at a line-of-fit on a scatter plot? Does that line have a positive slope? If so, your data shows a positive correlation! Learn about positive correlation by watching this tutorial.

#### What's Negative Correlation?

Looking at a line-of-fit on a scatter plot? Does that line have a negative slope? If so, your data shows a negative correlation! Learn about negative correlation by watching this tutorial.

#### What Does it Mean To Have No Correlation?

Scatter plots are very helpful in graphically showing the pattern in a set of data. But sometimes that data shows no correlation. Learn about no correlation and see how to tell if data shows no correlation by watching this tutorial!

#### What Does Negative Slope Mean?

What does a negative slope mean? What does the graph of a negative slope look like? Find the answers to these questions by watching this tutorial!

#### What Are Vertical Lines?

You may be able to guess that vertical lines are lines that go straight up and down, but did you know that all vertical lines have the same slope? In this tutorial, learn all about vertical lines including their slope and what the equation of a vertical line looks like!

#### What Are Horizontal Lines?

Ever look at the horizon when the sun is rising or setting? Know why it's called the horizon? It's a horizontal line! And just like the horizon, horizontal lines go straight left and right. In this tutorial, you'll learn all about horizontal lines including their slope and what the equation of a horizontal line looks like.

#### What Does Positive Slope Mean?

What does a positive slope mean? What does the graph of a positive slope look like? Find the answers to these questions by watching this tutorial!

#### What Does 0 Slope Mean?

A zero slope is just the slope of a horizontal line! The y-coordinate never changes no matter what the x-coordinate is! In this tutorial, learn about the meaning of zero slope.

#### What Does Undefined Slope Mean?

An undefined slope (or an infinitely large slope) is the slope of a vertical line! The x-coordinate never changes no matter what the y-coordinate is! There is no run! In this tutorial, learn about the meaning of undefined slope.

#### What's Point-Slope Form of a Linear Equation?

When you're learning about linear equations, you're bound to run into the point-slope form of a line. This form is quite useful in creating an equation of a line if you're given the slope and a point on the line. Watch this tutorial, and learn about the point-slope form of a line!

#### What's Slope-Intercept Form of a Linear Equation?

When you're learning about linear equations, you're bound to run into the point-slope form of a line. This form is quite useful in creating an equation of a line if you're given the slope and a point on the line. Watch this tutorial, and learn about the point-slope form of a line!

#### What's the Y-Intercept?

When you have a linear equation, the y-intercept is the point where the graph of the line crosses the y-axis. In this tutorial, learn about the y-intercept. Check it out!

#### How Do You Solve a Word Problem Using the Direct Variation Formula?

Word problems allow you to see math in action! Take a look at this word problem involving an object's weight on Earth compared to its weight on the Moon. See how the formula for direct variation plays an important role in finding the solution. Then use that formula to see how much you would weigh on the Moon!

#### What's the Direct Variation or Direct Proportionality Formula?

Ever heard of two things being directly proportional? Well, a good example is speed and distance. The bigger your speed, the farther you'll go over a given time period. So as one variable goes up, the other goes up too, and that's the idea of direct proportionality. But you can express direct proportionality using equations, and that's an important thing to do in algebra. See how to do that in the tutorial!

#### How Do You Use the Formula for Direct Variation?

If two things are directly proportional, you can bet that you'll need to use the formula for direct variation to solve! In this tutorial, you'll see how to use the formula for direct variation to find the constant of variation and then solve for your answer.

#### How Do You Find the X- and Y-Intercepts of a Line If You Have a Graph?

When you're dealing with graphs, it's often important to identify the x-intercept and y-intercept. In this tutorial, you'll see how to find the x-intercept and y-intercept of a line. Take a look!

#### How Do You Find the Rate of Change Between Two Points on a Graph?

The rate of change is a rate that describes how one quantity changes in relation to another quantity. In this tutorial, practice finding the rate of change using a graph. Check it out!

#### How Do You Find the Rate of Change Between Two Points in a Table?

The rate of change is a rate that describes how one quantity changes in relation to another quantity. This tutorial shows you how to use the information given in a table to find the rate of change between the values in the table. Take a look!

#### How Do You Write the Equation of a Line in Slope-Intercept Form If You Have a Graph?

Working with the graph of a line? Trying to find the equation for that graph? Just pick two points on the line and use them to find the equation. This tutorial shows you how to take two points on the graph of a line and use them to find the slope-intercept form of the line!

#### How Do You Write the Equation of a Line in Slope-Intercept Form If You Have a Table?

Looking at a table of values that represents a linear equation? Want to find that equation? Then check out this tutorial! You'll see how to use values from a table to find the slope-intercept form of the line described in the table.

#### How Do You Use Point-Slope Form to Write an Equation from a Table?

Looking at a table of values that represents a linear equation? Want to find that equation? Then check out this tutorial! You'll see how to use values from a table and the point-slope form of a line to find the slope-intercept form of the line described in the table.

#### How Do You Find the Slope of a Line If You Have a Parallel Line?

Looking at a graph of parallel lines? Got the equation of one of the lines? Want to find the slope of the other line? No problem! Just remember that parallel lines have the same slope! Use the given equation to find the slope of the first line and since the lines are parallel, that's the slope of the second line! To see an example, check out this tutorial.

#### How Do You Make a Scatter Plot?

Scatter plots are a very useful way to help you visually see data. In this tutorial, you'll see how to take data from a table and plot it to create a scatter plot. Take a look!

#### How Do You Write and Use a Prediction Equation?

Scatter plots are a great way to see data visually. They can also help you predict values! Follow along as this tutorial shows you how to draw a line of fit on a scatter plot and find the equation of that line in order to make a prediction based on the data already given!

#### What is Rate of Change?

Trying to describe the how something changes in relation to something else? Use rate of change! In this tutorial, learn about rate of change and see the difference between positive and negative rates of change!

## Lecture 1: The Geometry of Linear Equations

The first lecture starts with Gilbert Strang stating the **fundamental problem of linear algebra**, which is to solve systems of linear equations. He proceeds with an example.

The example is a system of two equations in two unknowns:

There are three ways to look at this system. The first is to look at it a row at a time, the second is to look a column at a time, and the third is use the matrix form.

If we look at this equation a row at a time, we have two independent equations 2x - y = 0 and -x + 2y = 3 . These are both line equations. If we plot them we get the row picture:

The row picture shows the two lines meeting at a single point (x=1, y=2). That's the solution of the system of equations. If the lines didn't intersect, there would have been no solution.

Now let's look at the columns. The column at the x's is (2, -1), the column at y's is (-1, 2) and the column at right hand side is (0, 3). We can write it down as following:

This is a linear combination of columns. What this tells us is that we have to combine the right amount of vector (2, -1) and vector (-1, 2) to produce the vector (0, 3) . We can plot the vectors in the column picture:

If we take one green x vector and two blue y vectors (in gray), we get the red vector. Therefore the solution is again (x=1, y=2).

The third way to look at this system entirely through matrices and use the matrix form of the equations. The matrix form in general is the following: A**x** = **b** where A is the coefficient matrix, **x** is the unknown vector and **b** is the right hand side vector.

How to solve the equations written in matrix form will be discussed in the next lectures. But I can tell you beforehand that the method is called Gauss elimination with back substitution.

For this two equations, two unknowns system, the matrix equation Ax=b looks like this:

The next example in the lecture is a system of three equations in three unknowns:

We can no longer plot it in two dimensions because there are three unknowns. This is going to be a 3D plot. Since the equations are linear in unknowns x, y, z, we are going to get three planes intersecting at a single point (if there is a solution).

Here is the row picture of 3 equations in 3 unknowns:

The red is the 2x - y = 0 plane. The green is the -x + 2y - z = -1 plane, and the blue is the -3y + 4z = 4 plane.

Notice how difficult it is to spot the point of intersection? Almost impossible! And all this of going one dimension higher. Imagine what happens if we go 4 or higher dimensions. (The intersection is at (x=0, y=0, z=1) and I marked it with a small white dot.)

The column picture is almost as difficult to understand as the row picture. Here it is for this system of 3 equations in 3 unknowns:

The first column (2, -1, 0) is red, the second column (-1, 2, -3) is green, the fourth column (0, -1, 4) is blue, and the result (0, -1, 4) is gray.

Again, it's pretty hard to visualize how to manipulate these vectors to produce the solution vector (0, -1, 4). But we are lucky in this particular example. Notice that if we take none of red vector, none of green vector and one of blue vector, we get the gray vector! That is, we didn't even need red and green vectors!

This is all still tricky, and gets much more complicated if we go to more equations with more unknowns. Therefore we need better methods for solving systems of equations than drawing plane or column pictures.

The lecture ends with several questions:

- Can A
**x**=**b**be solved for any**b**? - When do the linear combination of columns fill the whole space?,
- What's the method to solve 9 equations with 9 unknowns?

The examples I analyzed here are also carefully explained in the lecture, you're welcome to watch it:

Topics covered in lecture one:

- [00:20] Information on textbook and course website.
- [01:05] Fundamental problem of linear algebra: solve systems of linear equations.
- [01:15] Nice case: n equations, n unknowns.
- [02:20] Solving 2 equations with 2 unknowns
- [02:54] Coefficient matrix.
- [03:35] Matrix form of the equations. Ax=b.
- [04:20] Row picture of the equations - lines.
- [08:05] Solution (x=1, y=2) from the row picture.
- [08:40] Column picture of the equations - 2 dimensional vectors.
- [09:50] Linear combination of columns.
- [12:00] Solution from the column picture x=1, y=2.
- [12:05] Plotting the columns to produce the solution vector.
- [15:40] Solving 3 equations with 3 unknowns
- [16:46] Matrix form for this 3x3 equation.
- [17:30] Row picture - planes.
- [22:00] Column picture - 3 dim vectors.
- [24:00] Solution (x=0, y=0, z=1) from the column picture by noticing that z vector is equal to b vector.
- [28:10] Can Ax=b be solved for every b?
- [28:50] Do the linear combinations of columns fill the 3d space?
- [32:30] What if there are 9 equations and 9 unknowns?
- [36:00] How to multiply a matrix by a vector? Two ways.
- [36:40] Ax is a linear combination of columns of A.

Here are my notes of lecture one. Sorry about the handwriting. It seems that I hadn't written much at that time and the handwriting had gotten really bad. But it gets better with each new lecture. At lecture 5 and 6 it will be as good as it gets.

The next post is going to be about a systematic way to find a solution to a system of equations called elimination.

## 3.2 Literal equations

We often have to solve an equation for one of the variables in terms of the other

variables. What we are doing in such cases is essentially postponing the substitution of

the values of the variables to the last. Consider the following problem for example.

**Example 1:** A man wants to buy a new car priced at $18,000, financing it at 5.75%

per annum and paying back the loan in monthly installments for 5 years.

What is the monthly payments?

We could have substituted the values into the installment purchase formula

and obtained the equation

We can solve this equation as follows:

So, the monthly payment is $345.90.

Or we can substitute the values in the monthly payment formula

which is exactly the same as the value obtained above.

Let us review how we obtained the monthly payment formula from the installment

purchase formula.

We have to solve this equation for M. Adding to both sides of the equation, we get

Multiplying both sides of the equation by C and then dividing both sides of the equation

by B, we obtain

## Method 2: Use the slope and y-intercept

A linear equation written in the form (y = mx + b) is said to be in slope-intercept form. This form shows the slope (m) and the y-intercept (b) of the graph. Knowing these two values will let you quickly draw the graph of the linear equation, as you can see in the example below.

### Example

Graph the linear equation:

(y = dfrac<2><3>x + 4)

### Solution

Since this equation is in the form (y = mx + b), you know that:

Let’s look at the y-intercept first.

The y-intercept is the point where the graph crosses the y-axis (the vertical axis). So, you can plot this point as:

Now consider the slope. Slope can be viewed as a rate of change: it represents the change in (y) over the change in (x). Sometimes, this is called “rise over run”.

This can be translated to:

Therefore, to find another point on the line, start with the y-intercept and go 3 unit right and 2 units up. Do this again, and you will find another point. In fact, you can keep doing this to find as many points as you think you will need to sketch a good graph.

Connect the points, and you have the graph of your linear function!

This method for graphing linear equations can be applied even when the slope is negative or when the slope is not a fraction, even if it doesn’t seem that way. The next example will show you how that works!

### Example

Graph the linear equation:

(y = -2x + 1)

### Solution

The y-intercept here is 1, so plot this point first.

The slope is –2. While this isn’t a fraction, it can be viewed as one if you let the denominator equal 1.

Now this can be applied to find points on the graph.

Finally, connect the points to sketch the graph of the linear equation.

TIP: If the slope is in decimal form, see if you can convert it to a fraction to apply this method. Otherwise, method 1 might be best.

## Solving Systems of Linear Equations Using Substitution

A **system of** linear equations is just a set of two or more linear equations.

In two variables ( x &thinsp &thinsp and &thinsp &thinsp y ) , the graph of a system of two equations is a pair of lines in the plane.

There are three possibilities:

- The lines intersect at zero points. (The lines are parallel.)
- The lines intersect at exactly one point. (Most cases.)
- The lines intersect at infinitely many points. (The two equations represent the same line.)

**How to Solve a System Using** **The Substitution Method**

**Step 1 :**First, solve one linear equation for y in terms of x .**Step 2 :**Then substitute that expression for y in the other linear equation. You'll get an equation in x .**Step 3 :**Solve this, and you have the x -coordinate of the intersection.**Step 4 :**Then plug in x to either equation to find the corresponding y -coordinate.

**Note 1 :** If it's easier, you can start by solving an equation for x in terms of y , also &ndash same difference!

Solve the second equation for y .

Substitute 19 &minus 7 x for y in the first equation and solve for x .

3 x + 2 ( 19 &minus 7 x ) = 16 3 x + 38 &minus 14 x = 16 &minus 11 x = &minus 22 x = 2

Substitute 2 for x in y = 19 &minus 7 x and solve for y .

**Note 2 :** If the lines are parallel, your x -terms will cancel in step 2 , and you will get an impossible equation, something like 0 = 3 .

**Note 3 :** If the two equations represent the same line, everything will cancel in step 2 , and you will get a redundant equation, 0 = 0 .

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## Forms of linear equations

Linear equations can take on multiple forms including slope-intercept form, point-slope form, standard form, and more.

### Slope-intercept form

Slope-intercept form is probably the most commonly used form of a linear equation. It is typically expressed as

where m is the slope, and b is the y-intercept.

The equation of a line can be written in slope-intercept form when the slope and y-intercept of the line are known. Or, given an equation in slope-intercept form, it is easy to quickly identify the slope and y-intercept of the line. Graphing the equation of a line is also relatively simple when given an equation in slope-intercept form.

### Point-slope form

Point-slope form is similar to slope-intercept form except that it is based on some point on the line, rather than the y-intercept specifically. It is typically expressed as

where (x_{1}, y_{1}) represent a point on the line, and m is the slope. Like slope-intercept form, it is also useful for graphing, and has the benefit of being usable using any point on the line rather than just the y-intercept specifically, as is the case with slope-intercept form.

### Standard form

The standard form for the equation of a line is typically expressed as

Where A, B, and C are integers, and A and B are not equal to 0. One of the key benefits of standard form over slope-intercept or point-slope form is that it can be used to quickly find the x-intercept. It can also be used to find the y-intercept of a line, but this is something that is also relatively easy using either of the other forms. Once the x-intercept and y-intercept are known, standard form can also be used to graph the line, though it is slightly more tedious than using either of the other two mentioned forms.

Standard form is also the form of a linear equation that is typically used when solving systems of linear equations. Using either slope-intercept or point-slope form would make solving linear equations more difficult.

Linear equations are also expressed in other forms, but the above are some of the most common.

## Budgeting

A party planner has a limited budget for an upcoming event. She'll need to figure out how much it will cost her client to rent a space and pay per person for meals. If the cost of the rental space is $780 and the price per person for food is $9.75, a linear equation can be constructed to show the total cost, expressed as y, for any number of people in attendance, or x. The linear equation would be written as y = 9.75x + 780. With this equation, the party planner can substitute any number of party guests and give her client the actual cost of the event with the food and rental costs included.

## Graphical representation of Linear Equation

We can put the values of ‘x’ and ‘y’ into the equation in order to graph a linear equation. We can use the “intercept” points. Few below mentioned points must be follow:

- Put x = 0 into the equation and solve for y and plot the point (0,y) on the y-axis
- Put y = 0 into the equation and solve for x and plot the point (x,0) on the x-axis
- Finally, draw a straight line between the two points

**Check your skills to find the solutions of these linear equations:**