Applying Systems of Equations – Part 1

Honors Math – Grade 8

124

182

yx

yx

Let x represent the first number and y represent the second number. Translate each sentence into an algebraic equation.

The numbers are 5 and 8.

+The y variable is eliminated because 1 + -1 = 0

Solve the equation

2. Now substitute x = 5 in either equation and solve.

1. Write the equations in column form and add.

Twice one number added to another number is 18. Four times the first number minus the other number is 12. Find the numbers.1

Twice one # added to another is 18.

2x + y = 18

4 times the first minus the other is 12.

4x – y = 12

Define the Variables

224

132

ba

ba

Let a represent the first number and b represent the second number. Translate each sentence into an algebraic equation.

The numbers are -5 and 9.

-The a variable is eliminated because 2 – 2 = 0

Solve the equation

2. Now substitute a = -5 in either equation and solve.

1. Write the equations in column form; subtract

One number added to twice another number is 13. Four times the first number added to twice the other number is -2. What are the numbers?2

One # added to twice another is 13.

a + 2b = 13

4 times the first added to twice the other is -2.

4a + 2b = -2

Define the Variables

Van Adults StudentsTotal Cost

A 2 5 $77

B 2 7 $95

9572

7752

sa

sa

An adult ticket costs $16 and a student ticket costs $9.

-The a variable is eliminated because 2 – 2 = 0

Solve the equation

2. Now substitute s = 9 in either equation and solve.

1. Write the equations in column form; subtract

A youth group traveling in two vans visited Mammoth Cave in Kentucky. The number of people in each van and the total cost of a tour of the cave are shown in the table. Find the adult price and the student price of the tour.

3Let a = the cost for an adult ticket

and s = the cost of a student ticket.Define the Variables

Write a system of equations.

Adults + Students = TC2a + 5s = 772a + 7s = 95

94

9

ww

wgRich

Gannon made $6.5 million and

Charles Woodson made $2.5

million.

Substitute w + 4 for g in the first equation.

Group like terms

2. Now substitute w = 2.5 in either equation and solve.

One equation is solved for g; Substitute g= w+4

In 2003, Rich Gannon, the Oakland Raiders quarterback, earned $4 million more than Charles Woodson, the Raiders cornerback. Together they cost the Raiders approximately $9 million. How much did each make?

4Let g = Rich Gannon’s earnings & w

= Charles Woodson’s earnings.Define the Variables Write a

system of equations.

Together they cost the Raiders 9 million.

g + w = 9

Rich Gannon earned 4 million more than Woodson.

g = w + 4

4

9

wg

wg

Solve.

The Yankees won 26 World Series and the Reds won 5 World Series.

Substitute 5.2r for y in the first equation.

Group like terms

2. Now substitute r = 5 in either equation and solve.

One equation is solved for y; Substitute y = 5.2r

The New York Yankees and the Cincinnati Reds together have won a total of 31 World Series. The Yankees have won 5.2 times as many as the Reds. How many Worlds Series did each time win?

5Let y = Yankee wins and r

= Reds wins.Define the Variables Write a

system of equations.

Together they won a total of 31 World Series.

y + r = 31

The Yankees won 5.2 times as many as the Reds

y = 5.2r

ry

ry

2.5

31

Solve.

24

180

yx

yx

Angle X measures 102 degrees and Angle Y measures 78 degrees.

The x variable is eliminated because 1 + -1 = 0

2. Now substitute y=78 in either equation and solve.

1. Write the equations in column form and add.

Angles X and Y are supplementary and the difference between angle Y and angle X is -24. Find the angle measures.6

Let x = Angle X and y = Angle Y.

Define the Variables

Write a system of equations.

Supplementary angles are two angles whose sum is 180.

x + y = 180

The difference between Angle Y and Angle X is -24.

y – x = -24 or –x + y = -24

24

180

xy

yx(+)

Solve the equation

4.295

6.326

gb

gb

The statue is 15.6 feet tall

The g variable is eliminated because 1 + -1 = 0

2. Now substitute b=311 in either equation and solve

1. Write the equations in column form and add.

The total height of an office building and the granite statue that stands on top of it is 326.6 feet. The difference in heights between the building and the statue is 295.4 feet. How tall is the statue?7

Let b = the height of the building and let g = the height of the statue.

1

Define the Variables

Write a system of equations.

The total height of the building and the statue is 326.6

b + g = 326.6

The difference between them is 295.4

b – g = 295.4

4.295

6.326

gb

gb(+)

Solve the equation