# Setting up Equations

About a third of GRE math questions involve "algebra" - the process of substituting a letter for an unknown value and manipulating statements to solve a problem. One of the most important components of using algebra is knowing how to "set up" the initial equation; that is, to express the statement in terms of variables and mathematical operations.

## Rule

Solve for unknown elements by substituting a letter for an unknown variable and adjusting the equation accordingly.
For example, suppose a cheetah can run 4 miles per hour faster than twice a giraffe's top speed. If the giraffe's top speed is 20 miles per hour, how fast can the cheetah run?
Let c equal the cheetah's top speed and g equal the giraffe's top speed.
c = 2g + 4
                Take the initial equation.
c = 2(20) + 4
                Substitute the known value for g.
c = 40 +4
                Multiply out the parentheses.
c = 44
                Add the numbers. c is 44.

## Practice

Set up and solve the following word problems.
1. One slice of cake has 150 more calories than two slices of pie. If one slice of pie has 80 calories, how many calories does a slice of cake have?
2. Gary has worked at the auto shop 5 years longer than Pat, who has worked there 3 years less than Lindsay. If Lindsay has worked at the shop for 6 years, how many years has Gary worked there?
3. Jennifer has twice as many dachshunds as she has yorkies, and she has three fewer yorkies than poodles. If Jennifer has 8 dachshunds, how many poodles does she have?

## Comments

Most (but not all) questions which involve a story or sentences will need to be set up as algebraic equations. These so-called "word problems" are very common on the GRE. Typically the mathematical operations themselves are not hard; rather, the test tends to trip up test takers on setting the problem up the right way.

## Answers to Practice Questions

1. 310
Let c equal the calories in a slice of cake and p equal the number of calories in one slice of pie.
c = 2p + 150
                Take the initial equation.
c = 2(80) + 150
                Substitute a known value for p.
c = 160 + 150
                Work out the parentheses.
c = 310
                Add the values. A slice of cake has 310 calories.
2. 8 years
Let g equal the number of years Gary has worked, p equal the number of years Pat has worked, and l equal the number of years Lindsay has worked.
p = l - 3
                Take one of the initial equations.
p = (6) - 3
                Substitute the known value for l.
p = 3
                Subtract the values. Pat has worked for 3 years.
g = p + 5
                Take the other initial equation.
g = (3) + 5
                Substitute the known value for p.
g = 8
                Add the values. Gary has worked for 8 years.
3. 7
Let d equal the number of Jennifer's dachshunds, y equal the number of Jennifer's yorkies, and p equal the number of Jennifer's poodles.
2y = d
                Take the initial equation. d is twice y.
2y = 8
                Substitute the known value for d.
y = 4
                Divide both sides by 4. Jennifer has 4 yorkies.
p = y + 3
                Take the other initial equation.
p = (4) + 3
                Substitute the known value for y.
p = 7
                Add the values. Jennifer has 7 poodles.