[word problems part 3][section 28]

(1.) The sum of two numbers is 52. The difference of the

same two numbers is 20. Find the numbers.

x + y = 52

x - y = 20 here is the problem
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2x = 72 add equations
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2 2 divide each side by 2

x = 36 divide and cancel

36 + y = 52 replace x with 36

-36 -36 subtract 36 from each side
__________________
y = 16 subtract

result: x = 36 and y = 16

(2.) One number is 5 more than a second number. The sum of

the numbers is 115. What are the numbers?

y = x + 5

x + y = 115 here is the problem

-x -x subtract x from each side
____________________
y = -x + 115

y = x + 5 put this here
_____________________

2y = 120 add equations
__ ____
2 2 divide each side by 2

y = 60 divide and cancel

60 = x + 5 replace y with 60

-5 -5 subtract 5 from each side
________________
55 = x subtract

result: x = 55 and y = 60

(3.) One number is 3 times a second number. The larger

number decreased by the smaller is 18. Find the

numbers.

y = 3x

y - x = 18 here is the problem

+x +x add x to each side
________________
y = x + 18 add

3y = 3x + 54 multiply thru by 3

-y = -3x multiply equation 1 thru by -1
______________________
2y = 54 subtract equations
___ ___
2 2 divide each side by 2

y = 27 divide and cancel

27 = 3x replace y with 27
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3 3 divide each side by 3

9 = x divide and cancel

result: x = 9 and y = 27

(4.) The sum of two numbers is 21. One number is 3 less

than the other. Find the numbers.

x + y = 21

x = y - 3 here is the problem

-y -y subtract y from each side
____________________
x - y = -3 subtract

x + y = 21 put this here
__________________
2x = 18 add equations
___ ____
2 2 divide each side by 2

x = 9 divide and cancel

9 + y = 21 replace x with 9

-9 -9 subtract 9 from each side
_________________

y = 12 subtract

result: x = 9 and y = 12

(5.) The difference between two numbers is 4. Three times

the larger is 2 more than five times the smaller. Find

the numbers.

x - y = 4

3x = 5y + 2 here is the problem

-5y -5y subtract 5y from each side
____________________
3x - 5y = 2 subtract

-3x + 5y = -2 multiply thru by -1

5x - 5y = 20 multiply equation 1 thru by 5
_________________
2x = 18 add equations
___ ___
2 2 divide each side by 2

x = 9 divide and cancel

9 - y = 4 replace x with 9

-9 + y = -4 multiply thru by -1

+ 9 + 9 add 9 to each side
____________________
y = 5 add

result: x = 9 and y = 5

(6.) The sum of two numbers is 20. Twice the larger number

is three times the smaller number. Find the numbers.

x + y = 20

2y = 3x here is the problem

-3x -3x subtract 3x from each side
_____________________
-3x + 2y = 0 subtract

3x + 3y = 60 multiply equation 1 thru by 3
_________________
5y = 60 add equations
____ ___
5 5 divide each side by 5

y = 12 divide and cancel

x + 12 = 20 replace y with 12

-12 -12 subtract 12 from each side
________________
x = 8 subtract

result: x = 8 and y = 12

(7.) The width of a rectangle is 6 inches less than the

length. The perimeter is 72 inches. Find the length

and the width.

2L + 2w = 72

w = L - 6 here is the problem

-L -L subtract L from each side
________________________
-L + w = -6 subtract

-2L + 2w = -12 multiply thru by 2

2L + 2w = 72 put this here
__________________
4w = 60 add equations
___ ____
4 4 divide each side by 4

w = 15 divide and cancel

15 = L - 6 replace w with 12

+ 6 + 6 add 6 to each side
________________
21 = L add

result: w = 15 and L = 21

(8.) The length of a rectangle is 4 feet more than the

width. The perimeter is 40 feet. Find the length

and the width.

2L + 2w = 40

L = w + 4 here is the problem

-w -w subtract w from each side
__________________
L - w = 4 subtract

2L - 2w = 8 multiply thru by 2

2L + 2w = 40 put this here
_______________
4L = 48 add equations
___ ____
4 4 divide each side by 4

L = 12 divide and cancel

12 = w + 4 replace L with 12

-4 -4 subtract 4 from each side
_______________
8 = w subtract

result: w = 8 and L = 12

(9.) Two angles are supplementary. The measure of one angle

is 30^o more than twice the other. What is the measure

of each angle?

A + B = 180

B = 2A + 30 here is the problem

-2A -2A subtract 2A from each side
_______________________
-2A + B = 30 subtract

2A + 2B = 360 multiply equation 1 thru by 2
_________________________
3B = 390 add equations
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3 3 divide each side by 3

B = 130 divide and cancel

A + 130 = 180 replace B with 130

-130 -130 subtract 130 from each side
____________________
A = 50 subtract

result: A = 50 and B = 130

(10.) Two angles are complementary. The measure of one angle

is twice the measure of the other angle. What is the

measure of each angle?

A + B = 90

B = 2A here is the problem

-2A -2A subtract 2A from each side
_________________
-2A + B = 0 subtract

2A + 2B = 180 multiply equation 1 thru by 2
____________________
3B = 180 add equations
___ ____
3 3 divide each side by 3

B = 60 divide and cancel

60 = 2A replace B with 60
___ ____
2 2 divide each side by 2

30 = A divide and cancel

result: A = 30 and B = 60