Question

A system of equations is shown below

y = 3x - 1
y = - 2x + 4
what is the sum of X and Y in the solution to the system
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Answer 1

3

Explanation:

You want the value of (x+y) as determined by the system of equations ...

• y = 3x -1
• y = -2x +4

Solution

We can subtract the second equation from the first to get ...

(y) -(y) = (3x -1) -(-2x +4)

0 = 5x -5

0 = x -1

1 = x

Using the first equation to find y, we have ...

y = 3(1) -1 = 2

The sum of x and y is (x +y) = (1 +2) = 3.

Alternate solution

Let t = x+y. This means y = t -x.

Now, the equations become ...

• t -x = 3x -1
• t -x = -2x +4

Adding 4 times the second equation to the first gives ...

(t -x) +4(t -x) = (3x -1) +4(-2x +4)

5t -5x = -5x +15

Adding 5x and dividing by 5 gives ...

t = 3

The sum of x and y is 3.

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Additional comment

Sometimes you can find the value of the objective function directly, as in the second solution here.

The reason we chose 4 as a multiplier in the alternate solution is that we observed the equations could be written as ...

t -4x = -1

t +x = 4

where the variable x has coefficients with a ratio of -4. Using 4 as the multiplier eliminates the x-variable, leaving t — the variable whose value we want.

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