Solving System of Equations by Graphing

Key Concepts

• Write a system of equations
• Solve a system of equations approximately.

Introduction

Types of Systems 

The solution to a system of equations is the ordered pair (or pairs) common to all lines in the system when the system is graphed. 

y = –4x 

y = –2x + 8 

(–4, 16) is the solution to the system. 

Consistent system

If the lines intersect at exactly one point, the system has exactly one solution and is called a consistent system of equations. 

Inconsistent system

If the lines are parallel and do not intersect, the system has no solution and is called an inconsistent system. 

Dependent system

If the two equations are actually the same and graph the same line, the system has an infinite number of solutions and is called a dependent system. 

Write a system of equations 

Example 1: 

Ashley wants to use milk and orange juice to increase the amount of calcium and vitamin A in her daily diet. An ounce of milk contains 37 milligrams of calcium and 57 micrograms* of vitamin A. An ounce of orange juice contains 5 milligrams of calcium and 65 micrograms of vitamin A. How many ounces of milk and orange juice should Ashley drink each day to provide exactly 500 milligrams of calcium and 1,200 micrograms of vitamin A? 

Solution:  

Formulate: 

The first step in solving an application problem is to introduce the proper variables. Often, the question asked in the problem will guide you in this decision. Reading the last sentence in the example, we see that we must determine a certain number of ounces of milk and orange juice. So we introduce variables to represent these unknown quantities: 

x = Number of ounces of milk 

y = Number of ounces of orange juice 

37x + 5y = 500 

57x + 65y = 1,200 

Compute: 

Graph the system of equations. Find the point where the graphs intersect. 

Interpret: 

Drinking 12.5 ounces of milk and 7.5 ounces of orange juice each day will provide Ashley with the required amounts of calcium and vitamin A. 

Example 2: 

Justin wants to use cottage cheese and yogurt to increase the amount of protein and calcium in his daily diet. An ounce of cottage cheese contains 3 grams of protein and 15 milligrams of calcium. An ounce of yogurt contains 1 gram of protein and 41 milligrams of calcium. How many ounces of cottage cheese and yogurt should Justin eat each day to provide exactly 62 grams of protein and 760 milligrams of calcium? 

Solution: 

Formulate: 

x = Number of ounces of cottage cheese 

y = Number of ounces of yogurt 

3x + y = 62 

15x + 41y = 760 

Compute: 

Graph the system of equations. Find the point where the graphs intersect. 

Interpret: 

Eating 16.5 ounces of cottage cheese and 12.5 ounces of yogurt each day will provide Justin with exactly 62 grams of protein and 760 milligrams of calcium. 

Solve a system of equations approximately 

Example 3: 

What is the solution of the system of equations? 

y = 5x – 4  

y = –6x + 14 

Solution: 

Step 1: Use a graphing utility to graph both equations. Find the point of intersection. 

y = 5x – 4  

4.18 ≟ 5(1.64) – 4 

4.18 ≟ 8.2 – 4  

4.18 ≠ 4.2 

Set the expressions for y equal to each other and solve for x. 

5x – 4 = –6x + 14 

5x + 6x  = 14 + 4 

11 x = 18 

x = 18/11 

=1.63

Now substitute for x in either equation to find y. 

The exact solution is (18/11, 46/11).  

Exercise

• Sophia is painting her house. She can either buy Brand A paint and a paint roller tray or Brand B paint and a grid for the paint roller. For how many gallons of paint would the price for both options be the same? If Sophia needs 15 gallons of paint, which is the better option?

• Describe the solution set for the system of equations that includes the equation of the line shown and each equation below.

• y = ½ x – 3
• 2x + y = 5
• Write an equation in slope-intercept form that would have infinitely many solutions in a system of equations with 5x – 2y= 8.
• Roshaun has saved $150 and continues to add $10 each week. Keegan starts with $0 and saves $25 each week.
• In how many weeks will they have the same amount of money?
• What amount of money will they each have saved?
• Solve the following system of equations by graphing. Round your answers to the thousandths, if necessary.
  • y = 5x+1
  • y = 2x+6

• Solve each system of equations by graphing. Round your answers to the thousandths, if necessary.
  • y = -6x+5
  • y = 4x+3

• Solve each system of equations by graphing. Round your answers to the thousandths, if necessary.
  • y = 9x+2
  • y =-3x-4

• Solve each system of equations by graphing. Round your answers to the thousandths, if necessary.
  • y = 1/3x+9
  • y = -3/4x+4

• Use the graph to determine the solution for the system of equations.

• How does the graph show that the solution of the system of equations has an x-value between 2 and 3?
• What is the approximate solution of the system of equations?
• Gabriela is considering buying fleece jackets from Anastasia’s Monograms or Monograms Unlimited. Anastasia’s charges a one-time design fee and a price per jacket. Monograms Unlimited only charges a price per jacket.

Concept Map

What have we learned

• Write a system of equations based on the given scenario.
• Solve a system of equations approximately by substitution method.