A Guide to the GRE/Systems of Equations

Systems of Equations

Two equations with two variables can be solved by adjusting one equation so that one variable is expressed in terms of another, and substituting this value into the other equation.

In 3x + 3y = 15, x and y could conceivably have many values, from 1 and 4 to 7 and -2, and so on. Both variables can not be determined using this equation. However, if a second equation is added, it is possible.

3x + 3y = 15 These two are known as a “system of equations.”

x - y = 1 To solve, begin with the second equation.

x = y + 1 Phrase the second equation in terms of one variable. In this case, this is done by adding y to both sides.

3(y +1) + 3y = 15 Substitute this value into the other equation.

3y + 3 + 3y = 15 Expand the parentheses.

6y + 3 = 15 Combine the variables.

6y = 12 Subtract 3 from both sides.

y = 2 Divide both sides by 6. y is 2 (making x 3).

Practice

1. 5a - 3b = 21

2a + b = 15

If a and b satisfy the system of equations above, what is the value of a + b?

2. 2f - g = 2

    2g + 2f  = 20

If f and g satisfy the system of equations above, what is the value of f?

3. q + 4r = 21

    2r - q = 9

If q and r satisfy the system of equations above, what are the values of both q and r?

Answers to Practice Questions

1. 9

5a - 3b = 21

2a + b = 15 Take the second equation.

b = 15 - 2a Convert this equation to express one variable in terms of the other. In this case, subtract 2a from both sides.

5a - 3(15 - 2a) = 21 Substitute this value into the first equation.

5a - 45 + 6a = 21 Expand the parentheses.

11a - 45 = 21 Consolidate the variables.

11a = 66 Add 45 to both sides.

a = 6 Divide both sides by 11. a is 6, b is 3, and a + b = 9

2. 4

2g + 2f = 20 Take the second equation.

2g = 20 - 2f Convert this equation to express one variable in terms of the other. In this case, subtract 2f from both sides.

g = 10 - f Divide both sides by 2.

2f - (10 - f) Substitute this value into the first equation.

2f - 10 + f= 2 Expand the parentheses.

3f - 10 = 2 Consolidate the variables.

3f = 12 Add 10 to both sides.

f = 4 Divide both sides by 3. f is 4.

3. 5, 1

2r - q = 9 Take the second equation.

-q = 9 - 2r Convert this equation to express one variable in terms of the other. In this case, subtract 2r from both sides.

q = 2r - 9 Multiply both sides by -2

(2r - 9) + 4r = 21 Substitute this value into the first equation.

2r - 9 + 4r = 21 Expand the parentheses.

6r - 9 = 21 Combine the variables.

6r = 30 Add 9 to both sides. r is 5; q 1.