Lesson Objectives

- Demonstrate an understanding of how to graph a linear equation in two variables
- Learn how to solve a system of linear equations in two variables using graphing
- Learn how to identify a system of linear equations in two variables with no solution
- Learn how to identify a system of linear equations in two variables with an infinite number of solutions

## How to Solve a Linear System using the Graphing Method

Up to this point, we have only dealt with a single linear equation in two variables. Let's suppose we saw an equation such as:

x - y = -3

We can obtain the slope-intercept form of the line by solving for y:

y = x + 3

Now we can graph our equation by plotting the y-intercept (0,3) and using our slope (m = 1) to find additional points: In some cases, we will need to work with two linear equations in two variables at the same time. Let's suppose we also had the equation:

5x - y = 1

Again, we can obtain the slope-intercept form of the line by solving for y:

y = 5x - 1

Now we can graph our equation by plotting the y-intercept (0,-1) and using our slope (m = 5) to find additional points: We can combine these two equations into what is known as a "system of linear equations" or a "linear system". The solution for the system is any point (x,y), that satisfies both of the equations. When we look at the graph of any linear equation in two variables, each point on the line is a solution. To find the solution for a system using the graphing method, we graph each equation and look for the point (x,y) of intersection. This point of intersection will lie on both lines and, therefore, be a solution to our system. Let's graph our two equations together and look for the point of intersection.

x - y = -3

5x - y = 1 We can see the point of intersection occurs at (1,4). We can verify our solution by plugging in for x and y in each equation of the system.

x - y = -3

(1) - (4) = -3

-3 = -3

5x - y = 1

5(1) - (4) = 1

5 - 4 = 1

1 = 1

We can see that our point (1,4) works as a solution for both equations, therefore, it is the solution for the system.

Example 1: Solve each linear system using graphing.

7x + 3y = 9

2x + 3y = -6

Let's begin by placing each equation in slope-intercept form: $$y=-\frac{7}{3}x + 3$$ $$y=-\frac{2}{3}x - 2$$ Now, we can graph each equation and look for the point of intersection: We can see that the point of intersection occurs at (3,-4). Let's check the solution for the system in each original equation:

7(3) + 3(-4) = 9

21 - 12 = 9

9 = 9 ✔

2(3) + 3(-4) = -6

6 - 12 = -6

-6 = -6 ✔

Example 2: Solve each linear system using graphing.

2x - 3y = 9

5x + 3y = 12

Let's begin by placing each equation in slope-intercept form: $$y=\frac{2}{3}x - 3$$ $$y=-\frac{5}{3}x + 4$$ Now, we can graph each equation and look for the point of intersection: We can see that the point of intersection occurs at (3,-1). Let's check the solution for the system in each original equation:

2(3) - 3(-1) = 9

6 + 3 = 9

9 = 9 ✔

5(3) + 3(-1) = 12

15 + (-3) = 12

12 = 12 ✔

Example 3: Solve each linear system using graphing.

4x + 3y = 6

x + 3y = -3

Let's begin by placing each equation in slope-intercept form: $$y=-\frac{4}{3}x + 2$$ $$y=-\frac{1}{3}x - 1$$ Now, we can graph each equation and look for the point of intersection: We can see that the point of intersection occurs at (3,-2). Let's check the solution for the system in each original equation:

4(3) + 3(-2) = 6

12 + (-6) = 6

6 = 6 ✔

(3) + 3(-2) = -3

3 + (-6) = -3

-3 = -3 ✔

Example 4: Solve each linear system using graphing.

3x - 4y = -8

6x - 8y = 24

Let's begin by placing each equation in slope-intercept form: $$y=\frac{3}{4}x + 2$$ $$y=\frac{3}{4}x - 3$$ We can see that these two lines are parallel. They have the same slope (m = 3/4) and different y-intercepts. This means we will never be able to find a point of intersection or a solution for the system. Let's take a look at this graphically: We can clearly see that the two lines are parallel and will never intersect. We will state that there is no solution to our system.

Example 5: Solve each linear system using graphing.

-7x + 9y = -13

14x - 18y = 26

Let's begin by placing each equation in slope-intercept form: $$y=\frac{7}{9}x - \frac{13}{9}$$ $$y=\frac{7}{9}x - \frac{13}{9}$$ There is no need to graph anything here, as we can see that both equations of the system are the same. Again, when this occurs we have an infinite number of solutions.

x - y = -3

We can obtain the slope-intercept form of the line by solving for y:

y = x + 3

Now we can graph our equation by plotting the y-intercept (0,3) and using our slope (m = 1) to find additional points: In some cases, we will need to work with two linear equations in two variables at the same time. Let's suppose we also had the equation:

5x - y = 1

Again, we can obtain the slope-intercept form of the line by solving for y:

y = 5x - 1

Now we can graph our equation by plotting the y-intercept (0,-1) and using our slope (m = 5) to find additional points: We can combine these two equations into what is known as a "system of linear equations" or a "linear system". The solution for the system is any point (x,y), that satisfies both of the equations. When we look at the graph of any linear equation in two variables, each point on the line is a solution. To find the solution for a system using the graphing method, we graph each equation and look for the point (x,y) of intersection. This point of intersection will lie on both lines and, therefore, be a solution to our system. Let's graph our two equations together and look for the point of intersection.

x - y = -3

5x - y = 1 We can see the point of intersection occurs at (1,4). We can verify our solution by plugging in for x and y in each equation of the system.

x - y = -3

(1) - (4) = -3

-3 = -3

5x - y = 1

5(1) - (4) = 1

5 - 4 = 1

1 = 1

We can see that our point (1,4) works as a solution for both equations, therefore, it is the solution for the system.

### Solving a Linear System using Graphing

- Graph each equation of the system
- Identify the point of intersection
- Check the solution by plugging into each equation of the system

Example 1: Solve each linear system using graphing.

7x + 3y = 9

2x + 3y = -6

Let's begin by placing each equation in slope-intercept form: $$y=-\frac{7}{3}x + 3$$ $$y=-\frac{2}{3}x - 2$$ Now, we can graph each equation and look for the point of intersection: We can see that the point of intersection occurs at (3,-4). Let's check the solution for the system in each original equation:

7(3) + 3(-4) = 9

21 - 12 = 9

9 = 9 ✔

2(3) + 3(-4) = -6

6 - 12 = -6

-6 = -6 ✔

Example 2: Solve each linear system using graphing.

2x - 3y = 9

5x + 3y = 12

Let's begin by placing each equation in slope-intercept form: $$y=\frac{2}{3}x - 3$$ $$y=-\frac{5}{3}x + 4$$ Now, we can graph each equation and look for the point of intersection: We can see that the point of intersection occurs at (3,-1). Let's check the solution for the system in each original equation:

2(3) - 3(-1) = 9

6 + 3 = 9

9 = 9 ✔

5(3) + 3(-1) = 12

15 + (-3) = 12

12 = 12 ✔

Example 3: Solve each linear system using graphing.

4x + 3y = 6

x + 3y = -3

Let's begin by placing each equation in slope-intercept form: $$y=-\frac{4}{3}x + 2$$ $$y=-\frac{1}{3}x - 1$$ Now, we can graph each equation and look for the point of intersection: We can see that the point of intersection occurs at (3,-2). Let's check the solution for the system in each original equation:

4(3) + 3(-2) = 6

12 + (-6) = 6

6 = 6 ✔

(3) + 3(-2) = -3

3 + (-6) = -3

-3 = -3 ✔

### Systems of Linear Equations with No Solution

Up to this point, we have only seen linear systems with exactly one solution. When this occurs, we say our system is consistent, and the equations are independent. In some cases, we will not be able to obtain a solution to our linear system. This happens when the two lines are parallel. Recall that parallel lines have the same slope and will never intersect. Since there isn't a point that exists on both lines, there is no solution. When this occurs, our system is said to be "inconsistent". We can state our answer as "no solution" or use the empty set symbol "∅". Let's look at an example.Example 4: Solve each linear system using graphing.

3x - 4y = -8

6x - 8y = 24

Let's begin by placing each equation in slope-intercept form: $$y=\frac{3}{4}x + 2$$ $$y=\frac{3}{4}x - 3$$ We can see that these two lines are parallel. They have the same slope (m = 3/4) and different y-intercepts. This means we will never be able to find a point of intersection or a solution for the system. Let's take a look at this graphically: We can clearly see that the two lines are parallel and will never intersect. We will state that there is no solution to our system.

### Systems of Equations with Infinitely Many Solutions

Another special case scenario occurs when we see a system of equations that has an infinite number of solutions. When this happens, we will see the same equation algebraically manipulated to look different. Since the two equations are the same, what works as a solution to one, also works for the other. Therefore, there will be an infinite number of solutions, and the equations are said to be "dependent". We can state our answer as "infinitely many solutions". Let's look at an example.Example 5: Solve each linear system using graphing.

-7x + 9y = -13

14x - 18y = 26

Let's begin by placing each equation in slope-intercept form: $$y=\frac{7}{9}x - \frac{13}{9}$$ $$y=\frac{7}{9}x - \frac{13}{9}$$ There is no need to graph anything here, as we can see that both equations of the system are the same. Again, when this occurs we have an infinite number of solutions.

#### Skills Check:

Example #1

Solve each system using graphing. $$2x + 7y=49$$ $$15x + 7y=-42$$

Please choose the best answer.

A

$$(-1, 3)$$

B

$$(-7, 9)$$

C

$$(2, 5)$$

D

$$(0, -4)$$

E

$$(3, 7)$$

Example #2

Solve each system using graphing. $$x + y=-5$$ $$13x + y=7$$

Please choose the best answer.

A

$$(1, -6)$$

B

$$(5, 4)$$

C

$$(1, 9)$$

D

$$(-9, 8)$$

E

$$(7, 6)$$

Example #3

Solve each system using graphing. $$7x - 4y=16$$ $$5x + 4y=32$$

Please choose the best answer.

A

$$(8, -2)$$

B

$$(-1, 6)$$

C

$$(4, 3)$$

D

$$(7, -5)$$

E

$$(1, 6)$$

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