Algebra 1 Common Core Student Edition, Grade 8-9

What does the
solution represent in
the real w orld?
Check w h a t th e assigned
variables represent.
Here, (2 0 ,4 0 ) represents
20 large snack packs and
40 small snack packs.

How many solutions
can a system o f
lin ear equations
have?
A system can have
exactly one solution,
infinitely many solutions,
or no solution.

Using Syst em s of Equat i ons mUHEC RESPONSE

Snack Bar A snack bar sells two sizes of snack packs. A large snack pack is $5, and

a small snack pack is $3. In one day, the snack bar sold 60 snack packs for a total of

$220. How many small snack packs did the snack bar sell?

Step 1 W rite th e system of equations. Let x = th e n u m b e r of large $5 snack

packs, an d let y = th e n u m b e r of sm all $3 snack packs.

x + y = 60 Represent the total number of snack packs.

5x + 3 y = 220 Represent the amount earned from 60 snack packs.

Step 2 x + y — 6 0 Use th e fir s t e q u a tio n t o s o lve f o r y.

y = 60 - x Subtract x fro m each side.

Step 3 5x + 3(60 - x) = 220 Substitute 60 - x fo r y in the second equation.

5x + 180 — 3x = 220 Distributive Property

2x = 40 Simplify.

x=20 Divide each side by 2.

Step 4 20 + y = 60 S ubstitute 20 fo r x in the firs t equation.

y = 40 Subtract 20 from each side.

The system ’s solution is (20, 40). The snack bar sold 40 sm all snack packs.

4 0

* Got It? 3. You pay $22 to rent 6 video games. The store charges $4 for new gam es and

'i5 5 f $2 for older games. How m any new gam es did you rent?

If you get an identity, like 2 = 2, w h en you solve a system of equations, th e n th e system

has infinitely m any solutions. If you get a false statem ent, like 8 = 2, th e n th e system

has no solution.

I Syst ems With Infinitely M any Solut ions or No Solution

How many solutions does each system have?

0 1 = - 2y + 4 0 y = 3 x — 11

3.5x + 7y = 14 y - 3x = -1 3

Substitute — 2y + 4 for x in S ubstitute 3x — 11 for y in

3.5x + 7y = 14. y - 3 x = - 1 3.

3.5x + 7y = 14 y - 3 x = - 1 3

3 .5 (—2y + 4) + 7y = 14 ( 3 x - 1 1 )- 3 x = - 1 3

—7y + 14 + 7y = 14 - 1 1 = - 1 3 X

14 = 14 The system h as no solution.

The system has infinitely m any solutions.

Got It? 4. How m any solutions does th e system have? 6y + 5x = 8

2.5x + 3y = 4

374 Chapter 6 Systems of Linear Equations and Inequalities