Question

Solve by using elimination

x + 3y = 12
-5x + +y = -12

(Please explain how to get the answer if possible.)

Answer 1

(x, y) = (3, 3)

Explanation:

You want to solve by elimination the system of equations ...

• x +3y = 12
• -5x +y = -12

Elimination

The point of the elimination method is to combine the equations in a way that causes the coefficient of one of the variables to become zero. In general, this can be accomplished by multiplying each equation by the coefficient of the chosen variable in the other equation, then subtracting the results one from the other.

Considering the x-coefficients, we can multiply the first equation by -5, the second by 1 and subtract the first product from the second. This eliminates the x-variable.

1(-5x +y) -(-5)(x +3y) = 1(-12) -(-5)(12)

-5x +y +5x +15y = -12 +60 . . . . . . . . . . eliminate parentheses

16y = 48 . . . . . . . . . . . . . . . . . . . collect terms

y = 3 . . . . . . . . . . . . . . . . divide by 16

Complete the solution

Now, we need to find x. We can do this by substituting for y in either equation. We choose to use the first equation:

x + 3(3) = 12

x = 3 . . . . . . . . . . subtract 9

The solution to the system of equations is (x, y) = (3, 3).

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Additional comment

We chose to explain the elimination in terms of subtraction. That subtraction can be done in either order:

-5(equation 1) -1(equation 2)

or

1(equation 2) -(-5)(equation 1)

We chose the latter order so the coefficient of y would end up positive. We find fewer mistakes are made when the signs are positive.

Your curriculum materials may explain the elimination process in terms of addition. You may have noticed that subtracting -5 times the first equation is the same as adding 5 times the first equation. When you do this using addition, one of the multiplier coefficients needs to be the opposite of the coefficient of the variable in the other equation.

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Answer 2

Explanation:

what equations do we really have here ?

there seem to be typos in the equations.

I assume you truly have

x + 3y = 12

-5x + y = -12

now, elimination means that we multiply the equations (both sides to keep the equations in their information unchanged) by applying factors and then add the resulting equations.

the result should have "eliminated" one of the variables, so that we can solve for the remaining variable.

and then we use that result and one of the original equations to solve for the second variable.

in our case I suggest we try to eliminate x first.

so, we multiply the first equation by 5. the second equation can stay as it is.

and then we add both :

5x + 15y = 60

-5x + y = -12

-----------------------

0 16y = 48

y = 48/16 = 3

using e.g. the original first equation :

x + 3y = 12

we put in the already calculated value for y (3) and solve for x :

x + 3×3 = 12

x + 9 = 12

x = 12 - 9 = 3

x = 3

y = 3

you understand the principle ? please let me know, if you have further questions.

if your actual equations are different to my assumptions, you need to apply this in the same way as I showed you here to whatever the real equations are.