Algebra 1 - Solving Equations

One-Step Equations

100

Solve for b:

-7 = b - 3

b = -4

WORK
-7 = b - 3
-7 +3 = b - 3 +3
-4 = b

200

Solve for x:

x/3 = 2

x = 6

WORK
x/3 = 2
x/3 *3 = 2 *3
x = 6

300

Solve for x:

5x = 20

(When you see it written 5x that means the same thing as 5*x)

x = 4

WORK
5x = 20
5x/5 = 20/5
x = 4

400

Solve for m:

(4/5)*m = 28

m = 35

WORK
(4/5)*m = 28
(5/4)*(4/5)*m = 28*(5/4)
m = 35

500

Solve for p:

-2.54 = p + 7.17

p = -9.71

WORK
-2.54 = p + 7.17
-2.54- 7.17 = p + 7.17- 7.17
p = -9.71

Two-Step Equations

100

Solve for x:

2x + 3 = 15

(When you see it written 2x that means the same thing as 2*x)

x = 6

WORK
2x + 3 = 15
2x + 3 - 3 = 15 - 3
2x = 12
(2x)/2 = 12/2
x = 6

200

Solve for a:

(1/2)a + 5 = 18

a = 26

WORK
(1/2)a + 5 = 18
(1/2)a + 5 - 5 = 18 - 5
(1/2)a = 13
2*(1/2)a = 13*2
a = 26

300

Solve for t:

-t + 8 = 3

t = 5

WORK
-t + 8 = 3
-t + 8 - 8= 3 - 8
-t = -5 [which is the same thing as -1t = -5]
-1t/-1=-5/-1
t = 5

400

Solve for x:

(x – 7)/3 = -12

x = -29

WORK
(x – 7)/3 = -12
3*((x – 7)/3)= -12*3
x – 7 = -36
x – 7 + 7= -36 + 7
x = -29

500

Solve for x:

-24 = -10x + 3

x = 2.7

WORK
-24 = -10x + 3
-24 - 3= -10x + 3 - 3
-27 = -10x
-27/(-10)=-10x/(-10)
2.7 = x

Multi-Step Equations

100

Solve for m:

5 = 5m - 23 + 2m

m =4

WORK
5 = 5m - 23 + 2m
5 = 5m + 2m - 23
5 = 7m - 23
5 + 23 = 7m - 23 + 23
28 = 7m
28/7 = 7m/7
4 = m

200

Solve for x:

(x + 4) + 2x = 67

x = 21

WORK
(x + 4) + 2x = 67
x + 2x + 4 = 67 [addition is commutative]
3x + 4 = 67
3x + 4 - 4= 67 - 4
3x = 63
3x/3 = 63/3
x = 21

300

Solve for x:

-8(2x - 1) = 36

x = -7/4

WORK
-8(2x - 1) = 36
-16x + 8 = 36 [Distributive Property]
-16x + 8 - 8 = 36 - 8
-16x = 28
-16x/(-16) = 28/(-16)
x = -7/4

400

Solve for x:

4x -12 = -40

x =7

WORK

4x- 12 = -40 

     +12   +12 

4x= -28

4x/4 = -28/4 

x = -7 

500

Solve for x:

3x/4 - x/3 = 10

[remember what fractions really look like?]

x = 24

WORK
3x/4 - x/3 = 10
9x/12 - 4x/12 = 10 [write fractions/common denominator]
5x/12 = 10
12*(5x/12) = 10*12
5x/5 = 120/5
x = 24

Variables on Both Sides

100

Solve for b:

3b - 4 = b + 2

b = 3

WORK
3b - 4 = b + 2
3b - 4 + 4 = b + 2 + 4
3b = b + 6
3b - b = b - b + 6
2b = 6
2b/2 = 6/2
b = 3

200

Solve for x:

5x + 2 = 2x + 14

x = 4

WORK
5x + 2 = 2x + 14
5x + 2 - 2 2x + 14- 2
5x = 2x + 12
5x - 2x = 2x - 2x + 12
3x = 12
3x/3 = 12/3
x = 4

300

Solve for p:

1.5p = 1.25p + 8

p = 32

WORK
1.5p = 1.25p + 8
1.5p - 1.25p = 1.25p - 1/25p + 8
0.25p/0.25 = 8/0.25
p = 32

400

Solve for x:

2(5x - 1) = 3(x + 11)

x = 5

WORK
2(5x - 1) = 3(x + 11)
10x - 2 = 3x + 33 [Distribute]
10x - 3x - 2 = 3x - 3x + 33
7x - 2 + 2 = 33 + 2
7x/7 = 35/7
x = 5

500

Solve for x:

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

NO SOLUTION
-4 = 5

WORK
9m - 4 = -3m + 5 + 12m
9m - 4 = -3m + 12m + 5
9m -9m - 4 = 9m -9m + 5
-4 = 5 <~~~NO! So, No Solution!

Word Problems

100

Solve:

Five friends equally split a restaurant bill that comes to $32.50. How much does each friend pay?

Each friend pays $6.50.

WORK
5x = 32.50
5x/5 = 32.50/5
x = 6.5 ~~~> $6.50

200

Solve:

You work for 4 hours on Saturday and 8 hours on Sunday. You receive a $50 bonus. You earn $164.00 total. How much did you earn per hour?

You earned $9.50 per hour.

WORK
4x + 8x + 50 = 164
12x + 50 = 164
12x + 50 - 50 = 164 - 50
12x = 114
12x/12 = 114/12
x = 9.5 ~~~> $9.50

300

Solve:

Two buildings have the same total height. One building has 8 floors with height, h. The other building has a ground floor of 16ft and 6 other floors with height, h. Write and solve an equation to find the height, h, of these floors.

8h = 16 + 6h
h = 8

WORK
8h = 16 + 6h
8h - 6h = 16 + 6h - 6h
2h = 16
2h/2 = 16/2
h = 8

400

Solve

Online concert tickets cost $37 each, plus a service charge of $8.50 per ticket. The Web site also charges a transaction fee of $14.99 for the purchase. You paid $242.49. How many tickets did you buy?

You bought 5 tickets.

WORK
37t + 8.50t + 14.99 = 242.49
45.5t + 14.99 = 242.49
45.5t + 14.99 - 14.99 = 242.49 - 14.99
45.5t = 227.5
45.5t/45.5 = 227.5/45.5
t = 5 ~~~> 5 tickets

500

Solve

Pristine Printing will print business cards for $0.10 each plus a setup charge of $15. The Printing Place offers business cards for $0.15 each with a setup charge of $10. What number of business cards costs the same from either printer? BONUS: What would the cost be?

Printing 100 cards would cost the same from either business.

WORK
0.10n + 15 = 0.15n + 10
0.10n - 0.10n + 15 = 0.15n - 0.10n +10
15 = 0.05n + 10
15 - 10 = 0.05n + 10 - 10
5 = 0.05n
5/0.05 = 0.05n/0.05
100 = n ~~~>100 cards

BONUS answer: It would cost $25.00 to make 100 cards from either place.

WORK
0.10n + 15 = ??
0.10(100) + 15 = 25 ~~~>$25.00

0.15n + 10 = ??
0.15(100) + 10 = 25 ~~~>$25.00