7.1 Solving Systems of Linear Equations by Graphing

Leanne Thompson

7.1 Solving Systems of Linear Equations by Graphing

A system of linear equations consists of two or more linear equations grouped together. An example of a system of linear equations is shown below:

$2x+y = 7$
$x-2y = 6$

Solutions of a System of Linear Equations

A linear equation in two variables, such as $2x+y = 7$ , has an infinite number of solutions. Each point on the line would be considered a solution to the equation.

To solve a system of linear equations like the one above, we want to find the values of the variables that are solutions to both equations. In other words, we are looking for the ordered pair(s) (x, y) that make both equations true. These are called the solutions to a system of equations.

Table 7.1.1
Solutions to a System of Linear Equations
Solutions to a system of equations are the values of the variables that make all the equations true. A solution to a system of two linear equations is represented by an ordered pair (x, y).

To determine whether an ordered pair is a solution to a system of linear equations, we substitute the values of the variables into each equation. If the ordered pair makes both equations true, it is a solution to the system.

Example 1

Consider the system of equations below:

$3x - y = 5$
$x + y = 3$

Is the ordered pair (2, 1) a solution?

Table 7.1.2
Steps	Solution
Substitute x = 2 and y = 1 into both equations.	$3x - y = 5$ $3(2) - 1 = 5$ $6 - 1 = 5$ $5 = 5$ $x + y = 3$ $2 + 1 = 3$ $3 = 3$
Determine whether the ordered pair is a solution to the system of equations.	The ordered pair (2, 1) made both equations true; therefore, (2, 1) is a solution to this system.

Practice 1

Consider the system of equations below:

$x - y = -1$
$2x - y = -5$

Are the following ordered pairs a solution to this system of equations?

a) (–2, –1)

b) (–4, –3)

Number of Solutions to a System of Linear Equations

In this chapter, we will use three methods to solve a system of linear equations. The first method we’ll use is graphing.

The graph of a linear equation is a line. Each point on the line is considered to be a solution to the equation. For a system of two linear equations, we will graph two lines. The intersection point(s) of these lines will be considered the solution(s) to the system.

When we solve a system of two linear equations represented by a graph of two lines, there are three possible cases, as shown:


The lines intersect. Intersecting lines have one point in common. There is one solution to this system.	The lines are parallel. Parallel lines have no points in common. There is no solution to this system.	Both equations give the same line. Identical lines have infinitely many points in common. There are infinitely many solutions to this system.

Solve a System of Linear Equations by Graphing

Table 7.1.3
To solve a system of linear equations by graphing:
Step 1: Graph the first equation. Step 2: Graph the second equation. Step 3: Determine whether the lines intersect, are parallel, or are the same line. Step 4: Identify the solution to the system using the guidelines below: If the lines intersect, identify the point of intersection. Check to make sure it is a solution to both equations. This is the solution to the system. If the lines are parallel, the system has no solution. If the lines are the same, the system has an infinite number of solutions.

Example 2

Solve the given system by graphing. State how many solutions the system has.

$2x + y = 7$
$x - 2y = 6$

Table 7.1.4
Steps	Solution
Graph the first equation. Write the equation in slope y-intercept form, if necessary.	$2x + y = 7$ $y = -2x + 7$
Graph the second equation. Write the equation in slope y-intercept form, if necessary.	$x - 2y = 6$ $y = \dfrac{1}{2} - 3$
Determine whether the lines intersect, are parallel, or are the same line.	The lines intersect.
Identify the solution to the system using the guidelines below. Check to make sure it is a solution to both equations.	The lines intersect at (4, –1). $2(4) - 1 = 7$ $8 - 1 = 7$ $7 = 7$ $4 - 2(-1) = 6$ $4 + 2 = 6$ $6 = 6$ This system has one solution. The solution is (4, –1).

Practice 2

Solve the given system by graphing. State how many solutions the system has.

a) $y = -x + 5$
$y = 2x - 1$

b) $y = \dfrac{1}{2}x - 3$
$x - 2y = 4$

c) $y = 2x - 3$
$-6x + 3y = -9$

Blank coordinate plane. Both axes have a minimum of -6 and maximum of 6 and a scale of 1.

Application of a System of Linear Equations

Practice 3

Sondra is making 10 quarts of punch from fruit juice and club soda. The number of quarts of fruit juice is 4 times the number of quarts of club soda. The following system of equations can be used to represent the relationship between the amount of fruit juice and the amount of club soda:

$f + c = 10$
$f = 4c$

How many quarts of fruit juice and how many quarts of club soda does Sondra need? Use the grid below to find the answer.

Blank coordinate plane. Both axes have a minimum of 0 and maximum of 10 and a scale of 1.

Finding an Intersection Point on a Graphing Calculator

If you have a system of equations with one intersection point, you can find this point on your graphing calculator by using the Intersect feature.

Table 7.1.5
To find the solution to a system of linear equations with the Intersect feature:
Write each equation in slope y-intercept form. Press Y= Graph one equation in Y₁= and the other equation in Y₂= Press 2nd Press TRACE Select 5: intersect Use the left or right arrow key to move the cursor close to the intersection point of your two lines. Press ENTER Use the left or right arrow key to move the cursor close to the intersection point of your two lines. Press ENTER Press ENTER The x value and the y value of the solution will be displayed at the bottom of the graph.

Practice 4

Verify your answer for Practice 2 a) using your calculator.

Homework

For each system, are the ordered pairs below solutions to the system?
$3x + y = 0$
$x + 2y = -5$

a) (1, –3)

b) (0, 0)

$x - 3y = -8$
$-3x - y = 4$

c) (2, –2)

d) (–2, 2)
Solve the given system by graphing. State how many solutions the system has. Check your answer using your graphing calculator.

a) $y = -x + 1$
$y = -5x - 3$ b) $y = \dfrac{3}{4}x + 1$
$y = \dfrac{3}{4}x + 2$ c) $y = -\dfrac{5}{4}x - 2$
$y = -\dfrac{1}{4}x + 2$

d) $y = -\dfrac{1}{2} + 4$
$y = -\dfrac{1}{2} + 1$ e) $y = -x - 4$
$y = -3$ f) $y = -4x - 3$
$y = -4x - 3$
Solve the given system by graphing. State how many solutions the system has. Check your answer using your graphing calculator.

a) $x + 3y = -9$
$5x + 3y = 3$ b) $x + 4y = -12$
$2x + y = 4$ c) $y = -\dfrac{1}{4}x + 2$
$x + 4y = -8$

d) $x - 3y = -3$
$x + y = 5$ e) $y = -3x - 6$
$6x + 2y = -12$ f) $3x + y = -1$
$2x + y = 0$

g) $y = 3x - 1$
$6x - 2y = 6$ h) $y = \dfrac{1}{2}x - 4$
$2x - 4y = 16$ i) $2x + y = 10$
$-x + y = 1$
In a system of linear equations, the two equations have the same slope. Describe the possible number of solutions to the system.
In a system of linear equations, the two equations have the same intercepts. Describe the possible number of solutions to the system.
Manny is making 12 quarts of orange juice from concentrate and water. The number of quarts of water is 3 times the number of quarts of concentrate. The following system of equations can be used to represent the relationship between the amount of orange juice and the amount of water:
$x + y = 12$
$y = 3x$

a) What does x represent, and what does y represent?
b) How many quarts of concentrate and how many quarts of water does Manny need? Use the grid below to find the answer.
Enrique is making a party mix for his dad’s birthday that contains raisins and nuts. For each ounce of nuts, he uses twice the amount of raisins, and he is going to make 24 ounces of party mix. The following system of equations can be used to represent the relationship between the amount of raisins and nuts:
$y = 2x$
$x + y = 24$

a) What does x represent, and what does y represent?
b) How many ounces of nuts and how many ounces of raisins does he need to make 24 ounces of party mix? Use the grid below to find the solution.
Leo is planning his spring flower garden. He wants to plant tulip and daffodil bulbs. He will plant 6 times as many daffodil bulbs as tulip bulbs. If he wants to plant 350 bulbs, how many tulip bulbs and how many daffodil bulbs should he plant? Use your graphing calculator and the system provided below to find the solution:
$y = 6x$
$x + y = 350$
Won is making lemonade from concentrate. The number of quarts of water he needs is 4 times the number of quarts of concentrate. How many quarts of water and how many quarts of concentrate does Won need to make 100 quarts of lemonade? Use your graphing calculator and the system provided below to find the solution:
$y = 4x$
$x + y = 100$
Molly is making strawberry-infused water. For each ounce of strawberry juice, she uses three times as many ounces of water, and she going to make 64 ounces of strawberry infused water. Which of the following equation(s) would be part of the system of equations to represent this scenario? Circle all that apply.

a) $y = -3x$
b) $x + y = 3$
c) $y = 3x$
d) $x + y = 64$
A system of equations contains the following relations:
$3x - 2y = 4$
$y = \dfrac{3}{2}x - 2$ How many solutions does this system have?

a) Infinitely many solutions
b) No solutions
c) One solution
d) Impossible to tell

Answers

1.

a) Yes

b) No

c) No

d) Yes

2.

a) (–1, 2)	b) No solution	c) (–4, 3)
d) No solution	e) (–1, –3)	f) Infinitely many solutions

3.

a) (3, –4)	b) (4, –4)	c) No solution
d) (3, 2)	e) Infinitely many solutions	f) (–1, 2)
g) No solution	h) Infinitely many solutions	i) (3, 4)

4. There could be no solutions (parallel lines) or infinitely many solutions (same line).

5. If their intercepts are the same, they would have to be the same line, so there would be infinitely many solutions.

6. a) x represents the number of quarts of concentrate; $y$ represent the number of quarts of water.
b) 3 quarts of concentrate and 9 quarts of water

7. a) x represents the number of ounces of nuts; $y$ represents the number of ounces of raisins.
b)

8

ounces of nuts and

16

ounces of raisins

8.

50

tulip bulbs and

300

daffodil bulbs

9. 20 quarts of concentrate and 80 quarts of water

10. c and d

11. a

Attribution

Unless otherwise indicated, material on this page has been adapted from the following resource(s):

Berg, T. (2020). Intermediate algebra. BCcampus. https://collection.bccampus.ca/textbooks/intermediate-algebra-bccampus-412/, licensed under CC BY-NC-SA 4.0

Marecek, L., Anthony-Smith, M., & Honeycutt, M. (2020). Elementary algebra 2e. OpenStax. https://collection.bccampus.ca/textbooks/elementary-algebra-2e-openstax-106/, licensed under CC BY 4.0

License

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a) $y = -x + 1$ $y = -5x - 3$	b) $y = \dfrac{3}{4}x + 1$ $y = \dfrac{3}{4}x + 2$	c) $y = -\dfrac{5}{4}x - 2$ $y = -\dfrac{1}{4}x + 2$

d) $y = -\dfrac{1}{2} + 4$ $y = -\dfrac{1}{2} + 1$	e) $y = -x - 4$ $y = -3$	f) $y = -4x - 3$ $y = -4x - 3$

a) $x + 3y = -9$ $5x + 3y = 3$	b) $x + 4y = -12$ $2x + y = 4$	c) $y = -\dfrac{1}{4}x + 2$ $x + 4y = -8$

d) $x - 3y = -3$ $x + y = 5$	e) $y = -3x - 6$ $6x + 2y = -12$	f) $3x + y = -1$ $2x + y = 0$

g) $y = 3x - 1$ $6x - 2y = 6$	h) $y = \dfrac{1}{2}x - 4$ $2x - 4y = 16$	i) $2x + y = 10$ $-x + y = 1$