Algebra 1 Common Core Student Edition, Grade 8-9

Go t It? 1. W hat is th e solution of each system? Use elim ination.

a. 5x - 6y = - 3 2 b. -3 x - 3y = 9

3x + 6y = 48 3x - 4y = 5

H o w is t h is p r o b le m
similar to Problem 1?
In each p ro b le m , you are
looking fo r coefficients
of one variable that
are e ith e r th e sam e or
opposites. Here, th e
coefficients of s are the
same, so e lim in a te s.

Solving a System by Subtracting Equations

Multiple Choice The theater club sells a total of 101 tickets to its first play. A student

ticket costs $1. An adult ticket costs $2.50. Total ticket sales are $164. How many

student tickets were sold?

(3 D 25 G D 42 C O 59 C O 76

Define Let a = th e n u m b e r of ad u lt tickets sold.

Let s = th e n u m b e r of stu d en t tickets sold.

Relate total n u m b e r of tickets total ticket sales

Write a + s = 101 2.5a + s = 164

S te p 1 Eliminate one variable. Since the difference of the coefficients of s is 0, eliminate s.

a+ s = 101

2.5a + s = 164

-1.5a + 0 = -63 Subtract the equations.

a = 42 Solve for a.

S te p 2 Solve for th e elim inated variable. Use eith er equation.

a + s = 101 You can use the first equation.

42 + s = 101 Substitute 42 for a.

s = 59 Solve for s.

There w ere 59 stu d en t tickets sold. The correct answ er is C.

Check 42 is close to 40 an d 59 is close to 60. The total n u m b e r of tickets is

about 40 + 60 = 100, w hich is close to 101. The total sales are about

$2.50(40) + $60 = $160, w hich is close to $164. The solution is reasonable.

Go t It? 2. W ashing 2 cars an d 3 trucks takes 130 m in. W ashing 2 cars a n d 5 trucks

takes 190 m in. H ow long does it take to w ash each type of vehicle?

In P roblem s 1 a n d 2, a variable is elim inated b ecau se th e sum or difference of its

coefficients is zero. From th e M ultiplication P roperty of Equality, you know th a t you

can m ultiply each side of a n eq u a tio n to get a new eq u a tio n th a t is equivalent to the

original. That is, a + b = c is equivalent to d(a + b) = dc, or da + db = dc. Since

this is true, you can elim inate a variable by adding or subtracting, if you first m ultiply an

e q u a tio n by an app ro p riate num ber. You can prove th a t th e results are th e sam e simply

by substituting th e values for th e variables in th e original eq u atio n s to show th a t the

eq u atio n s are true.

f 1 Lesson 6-3 Solving Systems Using Elimination rs 379