Equations by Graphing

Introduction

System of Linear Equations

A system of linear equations consists of two or more linear equations. 

A solution of a system of linear equations in two variables is an ordered pair of numbers that is a solution of both equations in the system. 

Example 1: 

Determine whether (–3, 1) is a solution of the system. 

x – y = – 4 

2x + 10y = 4 

Solution: 

Replace x with –3 and y with 1 in both equations. 

First equation: –3 – 1 = – 4 (True) 

Second equation: 2(–3) + 10(1) = – 6 + 10 = 4 (True) 

Since the point (–3, 1) produces a true statement in both equations, it is a solution of the system. 

Since a solution of a system of equations is a solution common to both equations, it is also a point common to the graphs of both equations. 

To find the solution of a system of two linear equations, we graph the equations and see where the lines intersect. 

Solve a system of equations by graphing 

Example 2: 

Solve the system by graphing.  

2x – y = 6 (Equation 1) 

x + 3y = 10 (Equation 2) 

Solution: 

First, graph 2x – y = 6. 

Second, graph x + 3y = 10. 

The lines APPEAR to intersect at (4, 2). 

Although the solution to the system of equations appears to be (4, 2), you still need to check the answer by substituting x = 4 and y = 2 into the two equations. 

First equation: 

2(4) – 2 = 8 – 2 = 6      (True) 

Second equation: 

4 + 3(2) = 4 + 6 = 10    (True) 

The point (4, 2) checks, so it is the solution of the system. 

Graph systems of equations with infinitely many solutions or no solution 

Example 3: 

What is the solution of each system of equations? Use a graph to explain your answer. 

–x + 3y = 6 

3x – 9y = 9 

Solution: 

First, graph –x + 3y = 6. 

Second, graph 3x – 9y = 9. 

The lines APPEAR to be parallel. 

Although the lines appear to be parallel, we need to check their slopes. 

–x + 3y = 6      First equation 

3y = x + 6                 Add x to both sides. 

y = 1/3 x + 2              Divide both sides by 3. 

3x – 9y = 9    Second equation 

–9y = –3x + 9            Subtract 3x from both sides. 

y =    1/3 x – 1             Divide both sides by –9. 

Both lines have a slope of 1/3, so they are parallel and do not intersect. Hence, there is no solution to the system. 

Example 4: 

What is the solution of each system of equations? Use a graph to explain your answer. 

x = 3y – 1   

2x – 6y = –2 

Solution: 

First, graph x = 3y – 1. 

Second, graph 2x – 6y = –2. 

The lines APPEAR to be identical. 

Although the lines appear to be identical, we need to check that their slopes and y-intercepts are the same. 

x = 3y – 1 First equation 

3y = x + 1         Add 1 to both sides. 

y = 1/3 x + 1/3         Divide both sides by 3. 

2x – 6y = – 2            Second equation 

–6y = – 2x – 2       Subtract 2x from both sides. 

y = 1/3 x + 1/3        Divide both sides by -6. 

Any ordered pair that is a solution of one equation is a solution of the other. This means that the system has an infinite number of solutions. 

Exercise

1. A ______________________ consists of two or more linear equations.
2. Determine whether (4, 2) is a solution of the system.
3. Solve the system by graphing

y = x – 1 (Equation 1)

y = −x + 3 (Equation 2)

• Use a graph to solve the following system of equation.

3x + 2y = 9

2/3y = 3 – x

• Use a graph to solve the following system of equation.

y = 1/2x + 7

4x – 8y = 12

• Use a graph to solve the following system of equation.

y = x

y = 2x + 1

• Determine whether the system of equations shown in the graph has no solution or infinitely many solutions.

• Determine whether the system of equations shown in the graph has no solution or infinitely many solutions.

Concept Map

What have we learned

• Solving a system of linear equations by graphing.
• Graph systems of equations with infinitely many solutions or no solution.