[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
____________
___      ___
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
_____________________

__       ____
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
____  ___
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
__________________
___      ____
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
_________________
___      ___
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
____________________

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
_________________
____  ___
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
__________________
___ ____
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
________________

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
_______________
___     ____
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 30o 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
_________________________
___    ____
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
____________________