Algebra Using Two Variables

The sum of two numbers is 44, and the larger number is 2 more than the smaller number. What is the smaller number? If x = the smaller number and y = the larger number, then which of the following systems of equations represents the word problem?

x + y = 44 and y = x + 2

If the numerator of a fraction is increased by 3, the fraction becomes 3/4. If the denominator is decreased by 7, the fraction becomes 1. What is the numerator of the fraction?

9

The sum of the two digits of a number is 9. If the tens digit is one-half the units digit, what is the number? Let t = the tens digit, u = the units digit, and t + u = 9. Which of the following equations would complete the system?

t= 1/2u

The ratio of the numerator to the denominator of a fraction is 2 to 3. If both the numerator and the denominator are increased by 2, the fraction becomes 3/4. What is the original fraction? Which of the following systems of equations can be used to solve the problem?

3n = 2d and 4n + 8 = 3d + 6

If the numerator of a fraction is increased by 3, the fraction becomes 3/4. If the denominator is decreased by 7, the fraction becomes 1. Determine the original fraction. Which of the following equations represents "If the numerator of a fraction is increased by 3, the fraction becomes 3/4"? (Hint: cross products)

4n + 12 = 3d

The sum of the two digits of a number is 16. The number formed by reversing the digits is 18 more than the original number. Determine the original number. Let t = the tens digit, u = the units digit, and u + t = 16. Which of the following equations would complete the system?

9u - 9t = 18

Which method is not used to find the solution of a system of linear equations.

Guess and Check

John is twice as old as Mary. The sum of their ages is 21. How old is Mary? Let J = John's age and M = Mary's age. Select the system equations that represents the problem.

J - 2M = 0 and J + M = 21

The sum of Pete's and Sam's ages is 30. Five years ago, Pete was 3 times as old as Sam. How old is Sam? Let P = Pete's age, S = Sam's age, and P + S = 30. Which of the following equations would complete the system?

P - 5 = 3S - 15

The sum of two numbers is 95. If the larger number is increased by twice the smaller number, the result is 120. What is the larger number? If x = the smaller number and y = the larger number, then which of the following systems of equations represents the word problem?

x + y = 95 and 2x + y = 120