SOLVING MULTI STEP EQUATIONS WITH VARIABLES ON BOTH SIDES

Solve each of the following equation.

Problem 1 :

-3m - 5 = 5m - 13

Solution :

Subtract 5m from both sides.

-3m - 5 - 5m = 5m -13 - 5m

-8m - 5 = -13

Add 5 to both sides.

-8m - 5 + 5 = -13 + 5

-8m = -8

Divide both sides by -8.

m = 1

So, the solution of m is 1.

Problem 2 :

6(y + 7) = 2(y - 3)

Solution :

6(y + 7) = 2(y - 3)

By distributing 6 and 2, we get

6y + 42 = 2y - 6

Subtract 2y from both sides.

6y + 42 - 2y = 2y - 6 - 2y

4y + 42 = -6

Subtract 42 from both sides.

4y + 42 - 42 = -6 - 42

4y = -48

Divide both sides by 4.

4y/4 = -48/4

y = -12

So, the solution of y is -12.

Problem 3 :

10q + 3q + 5 = 2(q - 3)

Solution :

10q + 3q + 5 = 2(q - 3)

10q + 3q + 5 = 2q - 6

13q + 5 = 2q - 6

Subtract 2q from both sides.

13q + 5 - 2q = 2q - 6 - 2q

11q + 5 = -6

Subtract 5 from both sides.

11q + 5 - 5 = -6 - 5

11q = -11

Divide both sides by 11.

11q/11 = -11/11

q = -1

So, the solution of q is -1.

Problem 4 :

(1/3)(x + 6) = (2/3)(x - 9)

Solution :

(1/3)(x + 6) = (2/3)(x - 9)

By distributing 1/3 and 2/3, we get

(x + 6)/3 = (2x - 18)/3

Multiply by 3 on both sides,

x + 6 = 2x - 18

Subtract 6 from both sides.

x = 2x - 18 - 6

x = 2x - 24

Subtract 2x from both sides.

x - 2x  = 2x - 24 - 2x

-x = -24

x = 24

So, the solution of x is 24.

Problem 5 :

8k+12=2(k-3)

Solution :

8k + 12 = 2(k - 3)

8k + 12 = 2k - 6

Subtract 2k from both sides.

8k + 12 - 2k = 2k - 6 - 2k

6k + 12 = -6

Add 6 from both sides.

6k + 12 + 6 = -6 + 6

6k + 18 = 0

6k = -18

Divide both sides by 6.

6k/6 = (-18)/6

k = -3

So, the solution of k is -3.

Problem 6 :

A food truck sells salads for $6.50 each and drinks for $2 each. The food truck's revenue from selling a total of 209 salads and drinks in one day was $836.50. How many salads were sold that day ?

a)  77      b)  93      c)  99     d) 105

Solution :

Total number of salads and drinks = 209

Let x be the total number of salads. Then 209 - x be number of drinks.

Cost of each salad = $836.50

Cost of each drink = $2

6.50 x + 2 (209 - x) = 836.50

6.50x + 418 - 2x = 836.50

4.50x = 836.50 - 418

4.50x = 418.5

x = 418.5/4.50

x = 93

So, number of salads is 93.

Problem 7 :

b = 2.35 + 0.25x

c = 1.75 + 0.40x

In the equation above, b and c represent the price per pound in dollars of beef and chicken respectively, x weeks after July during last summer. What was the price per pound of beef when it was equal to price per pound of chicken ?

a)  $2.60     b)  $2.85      c) $2.95     d)  $3.35

Solution :

b = 2.35 + 0.25x -----(1)

c = 1.75 + 0.40x -----(2)

b = c

2.35 + 0.25x = 1.75 + 0.40x

2.35 - 1.75 = 0.40x - 0.25x

0.6 = 0.15x

x = 0.6/0.15

x = 4

Applying the value of x in (2), we get

c = 1.75 + 0.40(4)

= 1.75 + 1.6

c = 3.35

So, option d is correct.

Problem 8 :

l = 24 + 3.5m

One end of a spring is attached to a ceiling. When an object of mass m kilograms is attached to the other end of the spring, the string streches to a length of l centimeters as shown in the equation above. What is m when l is 73 ?

a)  14       b)  27.7      c)  23     d) 279.5

Solution :

l = 24 + 3.5m

l = 73

73 = 24 + 3.5m

73 - 24 = 3.5m

49 = 3.5 m

m = 49/3.5

m = 14

So, the value of m is 14, option a is correct.