# SOLVING A SYSTEM BY MULTIPLYING AND ADDING

Solving a system by multiplying and adding :

In some linear systems, neither variable can be eliminated by adding or subtracting the equations directly. In systems like these, you need to multiply one of the equations by a constant so that adding or subtracting the equations will eliminate one variable.

## Solving a system by multiplying and adding : Steps

Step 1 :

Decide which variable to be eliminated.

Step 2 :

Multiply one equation by a constant to make the coefficient same for the variable which has to be eliminated.

Step 3 :

After having multiplied one equation by constant, add or subtract to eliminate that variable and solve for the other variable.

Step 4 :

Substitute the value of the variable received in step 3 into one of the equations to find the value of the eliminated variable.

## Solving a system by multiplying and adding - Example

Question :

Solve the system of equations by multiplying and adding.

3x - 5y  =  -17

2x + 15y  =  7

Step 1 :

Let us eliminate the variable y in the given two equations.

3x - 5y  =  -17 -------- (1)

2x + 15y  =  7 -------- (2)

Step 2 :

To make the coefficient of y same in both the equations, multiply the first equation by 3.

(1)  3 ----- > 9x - 15y  =  -51 -------- (3)

In equations (2) and (3), the variable y is having the same coefficient, but having different signs.

Step 3 :

Add the equations (2) and (3) to eliminate the variable y.

Divide both sides by 11.

11x / 11  =  - 44 / 11

x  =  - 4

Step 4 :

Substitute the value of x into one of the equations to find the value of y.

3x - 5y  =  -17

3(-4) - 5y  =  -17

-12 - 5y  =  -17

(-12 - 5y) + 12  =  (-17) + 12

-12 - 5y + 12  =  -17 + 12

Simplify.

-5y  =  -5

Divide both sides by -5

-5y / (-5)  =  -5 / (-5)

y  =  1

Hence, the solution to the system is

(x, y)  =  (-4, 1)

After having gone through the stuff given above, we hope that the students would have understood "Solving system of equations by multiplying and adding".

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