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## Applying Systems of Equations – Part 1

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**Applying Systems of Equations – Part 1**Honors Math – Grade 8**1**Twice one number added to another number is 18. Four times the first number minus the other number is 12. Find the numbers. Let x represent the first number and y represent the second number. Translate each sentence into an algebraic equation. Define the Variables 1. Write the equations in column form and add. Twice one # added to another is 18. 2x + y = 18 4 times the first minus the other is 12. 4x – y = 12 The y variable is eliminated because 1 + -1 = 0 + Solve the equation 2. Now substitute x = 5in either equation and solve. The numbers are 5 and 8.**2**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? Let a represent the first number and b represent the second number. Translate each sentence into an algebraic equation. Define the Variables 1. Write the equations in column form; subtract One # added to twice another is 13. a + 2b = 13 4 times the first added to twice the other is -2. 4a + 2b = -2 The a variable is eliminated because 2 – 2 = 0 - Solve the equation 2. Now substitute a = -5in either equation and solve. The numbers are -5 and 9.**3**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. Define the Variables Let a = the cost for an adult ticket and s = the cost of a student ticket. 1. Write the equations in column form; subtract The a variable is eliminated because 2 – 2 = 0 - Write a system of equations. Solve the equation 2. Now substitute s = 9 in either equation and solve. An adult ticket costs $16 and a student ticket costs $9.**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? Define the Variables Let g = Rich Gannon’s earnings & w = Charles Woodson’s earnings. 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 One equation is solved for g; Substitute g= w+4 Substitute w + 4 for g in the first equation. Rich Gannon made $6.5 million and Charles Woodson made $2.5 million. Group like terms Solve. 2. Now substitute w = 2.5 in either equation and solve.**5**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? Define the Variables Let y = Yankee wins and r = Reds wins. 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 One equation is solved for y; Substitute y = 5.2r Substitute 5.2r for y in the first equation. Group like terms Solve. 2. Now substitute r = 5 in either equation and solve. The Yankees won 26 World Series and the Reds won 5 World Series.**6**Angles X and Y are supplementary and the difference between angle Y and angle X is -24. Find the angle measures. Define the Variables Let x = Angle X and y = Angle Y. 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 1. Write the equations in column form and add. The x variable is eliminated because 1 + -1 = 0 (+) Solve the equation 2. Now substitute y=78 in either equation and solve. Angle X measures 102 degrees and Angle Y measures 78 degrees.**7**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? Define the Variables Let b = the height of the building and let g = the height of the statue. 1 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 1. Write the equations in column form and add. The g variable is eliminated because 1 + -1 = 0 (+) Solve the equation 2. Now substitute b=311in either equation and solve The statue is 15.6 feet tall