Algebra 1 Common Core Student Edition, Grade 8-9

How do you get
started?
There are parentheses
on b o th sides o f th e
equation. So, remove the
parentheses using the
Distributive Property.

How can you tell how

many solutions an

e q u a tio n has?

If you elim inate the

variable in the process

of solving, the equation

is e ith e r an id e n tity w ith

infinitely many solutions

or an equation w ith no

solution.

Pr o b l em 3 Solving an Equation W ith Grouping Symbols

What is the solution of 2(5x - 1) = 3 (x + 11)?

2(5x — 1) = 3(x + 11)

Distributive Property

Subtract 3x from each side.

Simplify.

Add 2 to each side.

Simplify.

Divide each side by 7.

Simplify.

lOx — 2 — 3x + 33

lOx — 2 — 3x = 3x + 33 — 3x

7x - 2 = 33

7x — 2 + 2 = 33 + 2

7x = 35

7x

7 '

35

' 7

G o t It? 3. W hat is the solution of each equation?

a. 4(2y + 1) = 2(y - 13) b. 7(4 - a) = 3(a - 4)

An equation th a t is tru e for every possible value of th e variable is an id e n tity. For

example, x + l = x + l i s a n identity. An eq u a tio n has no solution if th ere is no

value of th e variable th at m akes the eq u a tio n true. The eq u a tio n x + 1 = x + 2 has

no solution.

Pr o b l em 4 Identities a n d Equations W ith N o Solution

What is the solution of each equation?

0 10x + 12 = 2(5x + 6)

lOx + 12 = 2(5x + 6)

lOx + 12 = lOx + 12 Distributive Property

Because lOx + 12 = lOx + 12 is always true, th ere are infinitely m any solutions of the

equation. The original equ atio n is an identity.

0 9m — 4 = — 3 m + 5 + 12m

9 m - 4 = -3 m + 5 + 12m

9m -4 = 9m + 5 Combine like terms.

9m - 4 - 9m = 9m + 5 - 9m Subtract 9m from each side.

—4 = 5 X Simplify.

Because - 4 ¥= 5, the original equ atio n has no solution.

& Got It? 4. W hat is th e solution of each equation?

a. 3(4h - 2) =-6 + 12h b. 2x + 7 = - 1 ( 3 - 2x)

104 Chapter 2 Solving Equations