In this section linear equations will be discussed in more detail and how to graph them. The first thing to realize about linear expressions is that they are of degree 1. This means the largest exponent is 1. Here are some examples of linear equations.

x = 4

y = 2

a = 4 + c

x = y + 1

y = 2 x –1

x + y = 4

3 x + 4 y = 9

4 a – 6 b = 5

 

Here are some examples of some that are not,

y = x ² + 2

 

y = -3 x ²

 

y = x ⁴ + 3 x ³ – 6 x ² - 3 x + 2

 

All to look for is the degree (highest exponent). In the first two (y = x ² + 2 and y = -3x ²) the degree is 2. But in the last example y = x ⁴ + 3 x ³ - 6x ² - 3 x + 2 , the degree is 4. This is all to be concerned with for now. If it is not of degree 1, it is not linear. In other words, if its degree is anything higher than 1 it falls into a different class that will be covered later.

Anything of degree 1 is called “linear”. This also means that the graph will be a line.

In order to be able to graph, one must be able to plot coordinates. In order to do that, know what the x and y-axis are. The x-axis is the main line that runs back and forth on any graph. The y-axis runs up and down. Where these two lines cross is called the origin.

y-axis

 

 

 

 

 

 

x-axis

 

 

 

 

 

 

 

 

 

 

 

Any point on a graph is labeled with two numbers. The first number is called the x-coordinate and the second is the y-coordinate. The x-coordinates start at the left as being negative and as they move to the right they become positive once they cross the y-axis. The y-axis is negative on bottom and positive on top. The origin has a coordinate of (0,0), remember that the first 0 is the x-coordinate and the second is the y-coordinate. Now here is a graph with some points to show some examples of graphing points.

 

 

 

(-2,6)

(2,3)

(0,0)

 

(5,-1)

(-3,-5)

 

 

 

 

In graphing equations or functions, look at what kind it is to start with. As explained earlier, determine if it is linear or not. If it is linear (or of degree 1) it is a line. Hence, they are “linear”. Now to graph it, get the equation in the correct form. The form that is recommend is the slope-intercept form. That means that y is by itself on one side of the equation. When they take on the form y = m x + b, where m is called the slope and b is called the y-intercept. Now slope is nothing more than how much the graph changes. The easiest way to think about slope is in the form of business graphs. Think about the following graphs like they would be seen in a business meeting.

 

 

A

 

 

 

 

 

 

 

 

B

 

 

 

 

 

 

 

 

 

C

 

 

 

 

 

 

 

 

D

 

 

 

 

 

 

 

 

 

If these were business graphs and someone asked, “Here are four different companies (A, B, C, and D) which one would you like to invest your money into?” The obvious answer would be “A”. Now that is the easiest way to understand graphs and slope. Slope is how much something is changing. The steeper the line is on the graph the more slope it would have. Following is a summary of the graphs and their slopes.

 

Company A one would assume is doing very well. Moving from left to right it keep getting higher and higher. On a graph, its slope is positive.

 

Graph B is a company that is not doing so well, therefore it has negative slope. That is, when moving from left to right it keeps getting lower and lower.

 

Graph C is a company that is not changing at all. That means that the slope would be 0.

 

Company D is really confusing. It says that it is going straight up. Now with common sense everyone knows that it is not possible for a company graph to go straight up. Therefore, the slope of that graph would be undefined, or more exactly, ¥(infinity).

 

Since slope is m in y = m x + b, look at the other part of the y = m x + b equation, the b. It is called slope-intercept form, so the other part would be the intercept. Intercept is a lot like the word intersection. In this case look at the y-intercept. That means, where the graph crosses the y-axis. The letter in the equation for this is the b. In the following graphs there is a line along with the point where it crosses the y-axis or y-intercept.

 

 

 

 

 

 

(0,2)

 

 

 

 

 

 

 

 

 

This graph has a y-intercept of 2, because it crosses the y-axis at the point (0, 2). Looking at the graph, notice that the slope is positive (Remember the business graphs?).

 

 

 

 

 

 

 

 

 

(0,-4)

 

 

 

 

 

Looking at this graph it crosses the y-axis at –4, thus making its’ y-intercept, or b, equal to –4. The slope in this graph is negative because from left to right it looks like it is going down.

 

One important thing to notice is that so far in the two graphs there have been 2 different y-intercepts, (0,2) and (0, -4). In these two points, along with every other y-intercepts, the x-value (first number in (x, y)) will always be 0.

 

Slope is represented by a number and not just being positive or negative. Slope can also be called the rate of change. The term “rate of change” is referring to slope. Slope is best defined by the following equation.

 

 

 

When people refer to this formula it is usually read as “slope is equal to rise over run.”

Rise simply means how much something is rising. Keep in mind that “rise” is referring to how much the elevation is changing, or the height. The word “run” would refer to the change back and forth. There is an illustration of that in a moment. First focus on how much something is rising. To do that, find two points that the line goes through.

 

 

 

 

(4,7)

 

 

(0,1)

 

 

 

 

 

 

 

 

 

 

Look at the two points, any two points along the line would work. The only condition was that the line had to go through those points. In the next graph look at the “rise”. That is how far it is changing height from one point to the other.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Count on the graph and see that the difference in the height, or y direction, between the two points, as shown in the graph, is 6. This is the rise. Next, look at the difference in the back and forth, or x, direction.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Notice that the distance covered back and forth is 4. This is the run. Applying the formula

gets

 

This should be the last step. Keep in mind that the slope can be simplified, therefore .

 

 

One more thing to consider is whether the slope of this graph is positive or negative. By thinking back to the business graphs and looking at this graph one can see that the slope is positive.

 

To write the equation of a line in slope-intercept form (y = m x + b) first find the slope, which is m, and in this case it is equal to 3/2. Then find the y-intercept, or b, in this case by looking at the graph and seeing where it crosses the y-axis, the y-intercept is 1. Substitute the m and b into the general equation results in,

.

 

 

 

This is the equation of the line in slope-intercept form.

 

Note- When writing the equation of a line, do not fill in numbers for x and y, only the m and b (slope and y-intercept). This is done because the x’s and y’s are always changing. When looking at a line, notice that x can equal 0, 1,2, or any other number, and the y value will change depending upon what the x value is. That is why the x and y are called variables. The slope of a straight line as well as where it crosses the y-axis will never change, and are therefore called constants.

 

 

Find the equation of the following line in slope intercept form.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

The first things to find are the slope and y-intercept. The y-intercept is relatively simple to find. By just looking at the graph, notice that it crosses the y-axis at –3. That makes b = - 3. In order to find the slope, find the rise and the run between any two points that the graph goes through.

 

Note- When choosing the two points to find the slope, chose two points where the line goes exactly through the intersection on the grid. In other words, make sure the line goes through that exact point.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

The two points used to determine the rise and run were (0,-3) and (4, -4). The rise is equal 1 and the run is equal to 4 between these two points. Therefore,

.

 

Notice that the slope in this case is negative (from the business graphs).

That would get a slope of .

 

Remember that slope is m in the slope intercept form, and as discussed earlier the y-intercept, or b, is –3.

By plugging into the formula for m and b results in

.

 

 

Here are some examples where information is given and one is asked to graph the line.

 

Example

Graph the following line.

 

 

First, see what the y intercept is. It is 6, since that is the b value, which means the line crosses the y-axis at 6. So point (0,6) is on the graph. Graph that point.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

With that point graphed, use the slope of 2/5 from that point. Start at that point (0, 6), rise 2 and run 5 to get to the next point (5,8).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Now there is an additional point that is on the line (5,8). With any two points one can graph a line. All to do from here is to draw a line that goes through those two points.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Answer!

This is the graph of .

 

 

Note- When doing the rise over run in the previous example, the run was to the right. Do this anytime that there is a positive slope. If one has a negative slope then run to the left after rising.

 

 

 

 

 

 

Example

Graph .

 

 

The –3 represents slope so the slope is -3. Now –3 is the same thing as . Since slope is usually thought of as rise over run, it helps to have it in the form of one number over another. Thinking about it like , means that rise is 3 and run is 1. The negative is used when graphing, do not assign it to the 3 on top but rather to the entire fraction. Slope is known and the y-intercept is –2. As discussed earlier if –2 is the y-intercept then point (0-2) is on the graph so put that point on the graph.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

From that point rise 3 and run 1. Keep in mind the slope is negative, so run to the left.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Here are two points, (0,-2) and (-1,1). All to do now is to draw a straight line between them, and the graph is complete.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Answer! The graph of y = -3x-2.

 

 

 

 

Example

Graph the line that has a slope of and goes through the point (-4, 2).

 

The first things to do is to graph the point, rise 1, and run 2 from it.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Starting at the point (-4,2) rising 1 and running 2 will be at the point (-2,3). Now that there are two points, draw a line between them.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Answer!

 

 

 

 

 

 

 

Example

Graph the line that goes through the two points (2, 1) and (3, -5).

 

The first thing to do is to graph the two points. Once these two points are graphed, draw a line between them.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Answer!

 

 

 

 

 

There are 2 other situations that need to be addressed here. They are graphs of lines like the ones before but 2 special cases of lines.

 

 

Example

Graph y = 3

 

 

This same equation can be written in the form y = 0x + 3 and mean the same thing. The best way to understand this problem is to think “Where on the graph is y = 3?” The points (-2,3), (0,3), and (7,3). All have y values of three. Notice in the equation that there is not an x, that means that it does not matter what x is. So basically y = 3, and x is anything. The graph of y = 3 is actually a horizontal or flat line going back and forth. This is the only way to have y = 3 everywhere along the line. If one were to plot the points discussed earlier and graph a line through them, would result in the following graph.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Answer!

 

 

 

 

 

Example

Graph x = -9 Look for when is x = -9. The answer will be a vertical line where y can be anything but the x has to be -9. Here is a graph of x = -9.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Answer!

 

The above definitions of slope and slope-intercept form are useful when graphing. It is also very useful in working these types of problems algebraically (without a graph).

 

 

 

 

 

 

Example

Find the equation, in slope-intercept form, of the line that goes through point (1,2) and has a slope of 3.

 

 

The first thing to realize is that the slope is 3. That means that in the slope-intercept form of a line, m = 3. Write that down.

 

y = 3 x + b

 

Find out the b (y-intercept) in this equation. To do this, use the point (1,2) that the line goes through. Let x = 1 and y = 2 just like it does in the point. Plug those values in for x and y in the equation.

 

2 = 3(1) + b

 

Now there is an equation to solve for b. Solve for b, by first multiplying the 3 and 1.

2 = 3 + b

 

Subtract 3 from both sides.

 

-1 = b b = -1.

 

Go back to the very first equation, y = 3x + b. Remember all that was needed to be in slope-intercept form was the value of b. Since b = -1, plug it in for b.

y = 3 x - 1

 

Answer!

 

 

 

 

 

Example

Find the equation, in slope-intercept form, of the line that goes through point (3,-2) and has a slope of -1.

 

Plug the value for slope (-1) into the slope-intercept formula.

 

y = -1x + b

 

Before trying to find b, simplify this formula by dropping the 1 in front of the x. It is understood there is a 1 there when being multiplied.

 

y = -x + b

 

Plug the 3 and –2 in for the x and y. Make sure not to get these two in the wrong order!

 

-2 = -(3) + b

 

Solve for b by first multiplying the 3 and the -.

 

-2 = -3 + b

 

Add 3 to both sides.

 

1 = b

 

Plug the b into the formula.

 

y = -x + 1

 

Answer!

 

 

 

 

 

 

Example

Find the equation, in slope-intercept form, of the line that goes through point (-2, -6) and has a slope of

Plug in the slope.

y = x + b

 

Plug in the –2 and –6 for x and y.

-6 = (-2) + b

 

Solve for b by first multiplying.

-6 = + b

 

Subtract from both sides.

 

= b

 

Plug this value for b in to the equation.

y = x -

 

Answer!

 

 

 

 

 

Earlier this formula for slope was used.

 

 

 

 

 

When talking about the rise, it is actually referring to how far it travels up and down, or in the y-direction. To find out how much something changes, subtract. Subtracting the amount one started with, from the amount ended up with, typically does this. This is done with two points to calculate slope. In other words, subtract the y values for rise and subtract the x values for run. Here is another formula for calculating the slope.

 

 

 

 

This is using sub notation (the little numbers by the y’s and x’s). The sub notation does not mean to multiply or anything like that, but is simply used to name the different variables. Knowing that, understand this formula as saying “The second y minus the first y over the second x minus the first x.” It does not matter which point is number 1 in sub notation or number 2. Be consistent in the fact that the second y must be with the second x and so on. Anytime subtracting, one is actually measuring a difference. This is denoted in math with the D(delta) symbol. Using this symbol, rewrite the formula.

 

 

 

 

This new formula is very useful because it allows the calculation of the slope without having to graph it and calculate the rise over the run. Here are a few examples applying it.

 

 

 

 

Example

Find the slope of the line that goes through the two points (1, 2) and (5, 7).

 

The first thing to do to keep organized is to name the two points. It does not matter how they are named. But once done, keep it throughout the entire problem. Go ahead and name (1, 2) as point 1, and (5, 7) as point 2.

Point 1 = (1,2)

 

Point 2 = (5,7)

The points are named so find out the variables used to calculate the slope. The variables are x₁ ,y₁ ,x₂ , and y₂ . x₁ is simply the x in the first point and so on.

 

x₁ = 1

y₁ = 2

x₂ = 5

y₂ = 7

 

Plug these values in for the variables in the slope equation. Make sure to keep the y’s on top.

 

 

 

 

Solve.

 

 

 

Answer!

 

 

 

 

 

 

 

Example

Find the slope of the line that goes through the two points (3,4) and (6,1).

Name the points and find the variables.

 

x₁= 3

y₁= 4

x₂= 6

y₂= 1

 

Plug into the formula.

 

 

 

Solve.

 

 

This will simplify.

 

-1

 

Answer!

 

 

 

 

 

 

Example

Find the slope of the line that goes through the two points (-1, 0) and (3, 8).

Name the points and find the variables.

 

x₁ = -1

y₁ = 0

x₂ = 3

y₂ = 8

 

Plug into the formula.

 

 

 

Solve.

 

 

Simplify.

 

 

Simplify.

 

2

 

Answer!

 

 

Example

Find the equation of the line that goes through the two points (2,-3) and (0,7).

 

In order to find the equation, one must know the slope and the y-intercept. Find the slope first by using the same method as before. Name the points and find the variables.

 

x₁ = 2

y₁ = -3

x₂ = 0

y₂ = 7

 

Plug into the formula.

 

 

Solve.

 

 

Simplify.

 

-5

 

This is the slope so plug it into the formula.

 

y = -5 x + b

 

To complete the equation, find b. Do this just like earlier, by plugging in one of the point values for x and y. It does not matter which point to plug in either point 1 or point 2. Decide which one will be easiest to work with and use it. The second point (0,7) will be easiest in this case so plug it in.

 

7 = -5(0) + b

 

Solve for b by first multiplying.

 

7 = 0 + b

 

Simplify.

 

7 = b

 

Now the slope is known as well as the y-intercept so plug both of them in.

 

y = -5x + 7

 

Answer!

 

 

 

Generally, the most useful form for a line is slope intercept form. Which is what has been used so far. Standard form is another form that many like to use. It is not as easy to use in graphing purposes but algebraically it can be easier in some circumstances.

 

Standard form of a line is

Ax + By = C.

 

Where A, B, and C are constants (numbers). For example in the equation 3x – 4y = 2, A = 3, B = - 4 and C = 2. Make sure when writing a line in standard form that A is a positive whole number. Work out the problems just like before and if standard form is asked for, convert it to standard form from slope intercept form for the last step.

 

 

 

 

Example

Write y = 2x –6 in standard form.

 

To go to standard form subtract 2x from both sides.

-2x + y = -6

 

The only thing left to do is to get the A term, -2, to be positive. To do this divide everything on both sides of the equation by a –1.

 

2x – y = 6

 

Answer!

 

 

 

 

Example

 

Write in standard form.

 

Instead of dealing with fractions the best thing to do is to find the least common multiple of all the denominators in the problem and multiply every term by that. In this case there is only one denominator (2) so multiply every term by 2. This will get rid of all the fractions that occur in an equation.

2y = -1x + 8

 

Simplify the –1 in front of the x.

2y = -x + 8

 

Add x to both sides.

x + 2y = 8

 

Answer!