Question

The sum of two numbers is 86, and their difference is 20. Find the two numbers

Answer 1

Answer: 53 and 33

Explanation:

Let be "x" the first number and "y" the second number.

Based on the information provided we can set up a system of equations:

`\left \{ {{x+y=86} \atop {x-y=20}} \right.`

Applying the Elimination method, we can add both equations and solve for the variable "x":

`\left \{ {{x+y=86} \atop {x-y=20}} \right.\\................\\2x=106\\\\x=(106)/(2)\\\\x=53`

Now we can substitute
`x=53` into one of the original equations and solve for the variable "y":

`53+y=86\\\\y=86-53\\\\y=33`

Answer 2

x = 53, y = 33

Explanation:

Let one of the numbers = x, & the other = y. Set the equations:

x + y = 86

x - y = 20

This is a system of equation, in which the variables must be true for both of them.

First, choose a equation to use. In this case, I will choose x + y = 86. Isolate the variable, x. Note the equal sign, what you do to one side, you do to the other. Subtract y from both sides.

x + y = 86

x + y (-y) = 86 (-y)

x = 86 - y

Now, note the other equation: x - y = 20. You now know that x = 86 - y. Plug in 86 - y for x in the "other" equation:

x - y = 20

(86 - y) - y = 20

Simplify. Combine like terms:

86 - y - y = 20

86 - 2y = 20

Isolate the variable, y. Note the equal sign, what you do to one side, you do to the other. Do the opposite of PEMDAS (Parenthesis, Exponents (& Roots), Multiplication, Division, Addition, Subtraction):

First, subtract 86 from both sides:

86 (-86) - 2y = 20 (-86)

-2y = 20 - 86

-2y = -66

Isolate the variable, y. Divide -2 from both sides:

(-2y)/-2 = (-66)/-2

Note: Before we go any further, I want to remind you of division & multiplication rules. There are 3 kinds, when you are multiplying two positive numbers (+/+), when you are multiplying a positive number with a negative number (+/-), and when you are multiplying a two negative numbers (-/-). Note the "table" in the bottom:

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(+/+) : Two positive numbers multiplied/divided with each other will always result in a positive answer:

Examples:

10/5 = 2

20/10 = 2

(+/-) : One positive number and one negative number multiplied/divided with each other will always result in a negative answer:

Examples:

20 x -5 = -100

50 x -3 = -150

(-/-) : Two negative numbers multiplied/divided with each other will always result in a positive answer:

Examples:

-30 x -5 = 150

-200/-10 = 20

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Now, solve the equation:

(-2y)/-2 = (-66)/-2

y = -66/-2

Divide.

y = 33

One of your numbers (y) is 33. Plug in 33 for y in one of the equations:

x + y = 86

x + (33) = 86

Isolate the variable, x. Note the equal sign, what you do to one side, you do to the other. Subtract 33 from both sides.

x + 33 (-33) = 86 (-33)

x = 86 - 33

x = 53

Your answers: x = 53, y = 33.

Check: Plug in 53 for x, 33 for y in the equations:

x + y = 86:

(53) + (33) = 86

86 = 86 (True)

x - y = 20:

(53) - (33) = 20

20 = 20 (True)

Your answers: x = 53, y = 33

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