Lesson Objectives
• Demonstrate an understanding of Linear Equations in Two Variables
• 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 the Graphing Method
• Learn how to check the solution for a System of Linear Equations in Two Variables

How to Solve a System of Linear Equations in Two Variables using Graphing

At this point, you should be very comfortable with all aspects of working with a linear equation in two variables. So far, we have learned how to find ordered pair solutions, convert between different forms of a line, and quickly graph our equation using slope-intercept form. In this lesson, we will expand on our knowledge and learn how to solve a system of linear equations in two variables using the graphing method. We should understand that a linear equation in two variables has an unlimited number of solutions. Let's suppose we saw the following equation:
3x + 7y = 56
We could choose values for x and solve for y and create an infinite number of ordered pair solutions. A few solutions are:
(0, 8)
(7,5)
(-7,11)
What happens if we introduce another equation into the mix? Let's suppose we now saw the two equations:
3x + 7y = 56
8x - 7y = 21
Together, these two equations represent a system of linear equations in two variables. A system of linear equations will generally not have an infinite number of solutions. To solve a system of linear equations, we need to find the ordered pair that satisfies both equations of the system. Let's think about our above example:
3x + 7y = 56
8x - 7y = 21
If we use the ordered pair (7,5), we will see it works as a solution to each equation. We can check this by plugging in a 7 for each x and a 5 for each y:
3(7) + 7(5) = 56
21 + 35 = 56
56 = 56
8(7) - 7(5) = 21
56 - 35 = 21
21 = 21
In each case, the left and right sides are the same value. This ordered pair satisfies both equations of the system, therefore, it is the solution for our system.
There are many ways to solve a linear system. We can use graphing, substitution, elimination, or matrix methods. Each of these methods has its advantage, but graphing is the least practical. Graphing is tedious and not useful at all when non-integers are involved. It also doesn't work well with very small or very large values, as our handmade graphs can only reach so many numbers. Graphing is taught as a way to visualize the concept of solving a system of linear equations in two variables.
How can we solve a linear system using graphing? We graph each equation and then look for the point of intersection. Let's take a look at an example.
Example 1: Solve each linear system using graphing.
x + y = 2
4x + y = -1
We can easily graph each equation by placing each into slope-intercept form:
y = -x + 2
y = -4x - 1
Let's graph each equation and look for the point of intersection: We can see from our graph above that the point of intersection occurs at (-1,3). We can check this solution by plugging into each equation of the system:
-1 + 3 = 2
2 = 2
4(-1) + 3 = -1
-4 + 3 = -1
-1 = -1
Our solution checks out in each equation of the system, therefore, it is the solution for our system. Let's try another one.
Example 2: Solve each linear system using graphing.
3x + 7y = 21
12x + 7y = -42
We can easily graph each equation by placing each into slope-intercept form:
y = -3/7 x + 3
y = -12/7 x - 6
Let's graph each equation and look for the point of intersection: We can see from our graph that the point of intersection occurs at (-7,6). We can check this solution by plugging into each equation of the system:
3(-7) + 7(6) = 21
-21 + 42 = 21
21 = 21
12(-7) + 7(6) = -42
-84 + 42 = -42
-42 = -42
Our solution checks out in each equation of the system, therefore, it is a solution for our system.

Inconsistent Systems & Dependent Equations

We will face three types of linear systems:
1. Consistent Systems - Our system has one solution, the equations are said to be independent
2. Inconsistent Systems - Our system has no solution, the equations are parallel lines
3. Dependent Equations - Our system has an infinite number of solutions, the equations are the same
Our earlier examples contained examples of consistent systems. They had exactly one solution. In some cases, we will have no solution. This situation is known as an inconsistent system. This will occur whenever we have two parallel lines. Since parallel lines have the same slope, they will never intersect. This means there will not be a point that lies on both lines and therefore, there will never be a solution for the system. Let's look at an example.
Example 3: Solve each linear system using graphing.
3x + y = -4
9x + 3y = 3
We can easily graph each equation by placing each into slope-intercept form:
y = -3x - 4
y = -3x + 1
Let's graph each and look for a point of intersection: We can see that we have two parallel lines. The slopes are the same, and we have different y-intercepts. We will not have a solution for this system. When this situation occurs, we simply state "no solution".
Lastly, we will see dependent equations. This occurs when we have the same two equations in a system. Let's look at an example.
Example 4: Solve each linear system using graphing.
20x - 14y = 26
10x - 7y = 13
Let's solve each for y:
y = 10/7 x - 13/7
y = 10/7 x - 13/7
We have the same equation in each case, if we take the bottom equation and multiply each side by 2, we will obtain the top equation. For this scenario, we will say there is an infinite number of solutions.

Skills Check:

Example #1

Solve each system by graphing. $$x - y=-3$$ $$2x + y=-3$$

A
$$(-2, 1)$$
B
$$(4, 3)$$
C
$$(2, 5)$$
D
$$(3, -9)$$
E
$$(-1, 4)$$

Example #2

Solve each system by graphing. $$x - 4y=16$$ $$3x + 2y=6$$

A
$$(1, -3)$$
B
$$(-5, -7)$$
C
$$(4, -3)$$
D
$$(20, 1)$$
E
$$(5, 9)$$

Example #3

Solve each system by graphing. $$x - y=3$$ $$2x + y=3$$

A
$$(4, 1)$$
B
$$(2, -1)$$
C
$$(5, -7)$$
D
$$(4, 4)$$
E
$$\text{No Solution}$$