SAT MATH - SOLVING EQUATIONS WITH VARIABLES ON BOTH SIDES

Example 1 :

Solve for k :

5 - 3(k + 1) = -k

Solution :

5 - 3(k + 1) = -k

5 - 3k - 3 = -k

-3k + 2 = -k

Add k to both sides.

-2k + 2 = 0

Subtract 2 from both sides.

-2k = -2

Divide both sides by -2.

k = 1

Example 2 :

Solve for x :

Solution :

7x - 24 = 18 + x

6x - 24 = 18

6x = 42

x = 7

Example 3 :

7n + 3 = 2n - 12

Given the equation above, find the value of (-n + 3).

Solution:

7n + 3 = 2n - 12

5n + 3 = -12

5n = -15

n = -5

Multiply both sides by -1.

-n = 5

Add 3 to both sides.

-n + 3 = 8

Example 4 :

7(h - 5) - 3h = (³⁄₂)h

Given the equation above, find the value of (⅐)h.

Solution:

7(h - 5) - 3h = (³⁄₂)h

Multiply both sides by 2.

2[7(h - 5) - 3h] = 2(³⁄₂)h

14(h - 5) - 6h = 3h

14h - 70 - 6h = 3h

8h - 70 = 3h

5h - 70 = 0

5h = 70

h = 14

Multiply both sides by ⅐.

(⅐)h = ⅐ ⋅ 14

(⅐)h = 2

Example 5 :

Solution:

Substitute s = 2.

Example 6 :

⁽⁹ ⁻ ²ᵏ⁾⁄₃ = k - 2

Given the equation above, find the value of k.

Solution :

⁽⁹ ⁻ ²ᵏ⁾⁄₃ = k - 2

Multiply both sides by 3.

9 - 2k = 3(k - 2)

9 - 2k = 3k - 6

Subtract 3k from both sides.

9 - 5k = -6

Subtract 9 from both sides.

-5k = -15

Divide both sides by -5.

k = 3

Example 7 :

Four times the sum of three and a number equals nine less than the number.

(a) Write an equation for the problem.

(b) Then solve the equation.

Solution :

Part (a) :

Let x be the number.

Given : Four times the sum of three and a number equals nine less than the number.

4(x + 3) = x - 9

Part (b) :

Solve the equation in part (a).

4(x + 3) = x - 9

4x + 12 = x - 9

Subtract x from both sides.

3x + 12 = -9

Subtract 12 from both sides.

3x = -21

Divide both sides by 3.

x = -7

Example 8 :

A shirt costs $22 more than one half the cost of a pair of pants. If the cost of a shirt is $48, find the cost of a pair of pants.

Solution :

Let s be the cost of a shirt and p be the cost of a pair of pants.

Given :  The shirt costs $22 more than one half the cost of a pair of pants, you can set up the following equation.

Given : The cost of a shirt is $48.

Substitute s = 48.

Therefore, the cost of a pair of pants is $52.

Example 9 :

Twice a number n, increased by 11 is the same as six times the number decreased by 9. What is the value of n?

Solution :

From the given information,

2n + 11 = 6n - 9

Subtract 6n from from both sides.

-4n + 11 = -9

Subtract 11 from both sides.

-4n = -20

Divide both sides by -4.

n = 5

Example 10 :

One half of a number increased by 3 is five less than two thirds of the number. Find the number.

Solution :

Let x be the number.

From the given information,

Example 11 :

Four times the greatest of three consecutive odd integers exceeds three times the least by 31. What is the greatest of the three consecutive odd integers?

Solution :

Let x be odd integer. Then the three consecutive odd integers are

x, (x + 2), (x + 4)

Given : Four times the greatest of three consecutive odd integers exceeds three times the least by 31.

So, 4(x + 6) eceeds the 3x by 31.

For example, if 3x = 100, 4(x + 4) = 131.

131 = 100 + 31

Then, we have

4(x + 4) = 3x + 31

4x + 16 = 3x + 31

x + 16 = 31

x = 15

The greatest of the three consecutive odd integers :

= x + 4

= 15 + 4

= 19

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