A first number plus twice a second number is 10. Twice the first number plus the second totals 35. F

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A first number plus twice a second number is 10. Twice the first number plus the second totals 35. Find the numbers.

The phrase a number means an arbitrary variable
A first number is written as x
A second number is written as y

Twice a second number means we multiply y by 2:
2y

A first number plus twice a second number:
x + 2y

A first number plus twice a second number is 10 means we set x + 2y equal to 10:
x + 2y = 10

Twice the first number means we multiply x by 2:
2x

Twice the first number plus the second:
2x + y

Twice the first number plus the second totals 35 means we set 2x + y equal to 35:
2x + y = 35

Therefore, we have a system of two equations:

1. x + 2y = 10
2. 2x + y = 35

Since we have an easy multiple of 2 for the x variable, we can solve this by multiply the first equation by -2:

1. -2x - 4y = -20
2. 2x + y = 35

Because the x variables are opposites, we can add both equations together:
(-2 + 2)x + (-4 + 1)y = -20 + 35

The x terms cancel, so we have:
-3y = 15

To solve this equation for y, we type it in our search engine and we get:
y = -5

Now we substitute this y = -5 into equation 2:
2x - 5 = 35

To solve this equation for x, we type it in our search engine and we get:
x = 20