SOLVING A SYSTEM BY MULTIPLYING AND SUBTRACTING

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

The following steps will be useful to solve a system of equation using multiplication and subtraction. 

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. 

Question :

Solve the system of equations by multiplying and adding.

3x + 5y  =  11

2x + 15y  =  19

Answer :

Step 1 :

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

3x + 5y  =  11 -----(1)

2x + 15y  =  19 -----(2)

Step 2 :

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

(1) ⋅ 3 -----> 9x + 15y  =  33 -----(3)

In equations (2) and (3), the variable y is having the same coefficient, and also having the same sign.

Step 3 :

Subtract equation (2) from (3) to eliminate the variable y. 

(3) - (2) ----> (9x + 15y) - (2x + 15y)  =  33 - 19

9x + 15y - 2x - 15y  =  33 - 19

Simplify. 

7x  =  14

Divide both sides by 7. 

7x / 7  =  14 / 7

x  =  2

Step 4 : 

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

3x + 5y  =  11

3(2) + 5y  =  11

6 + 5y  =  11

Subtract 6 to both sides.

(6 + 5y) - 6  =  (11) - 6

6 + 5y - 6  =  11 - 6

Simplify.

5y  =  5

Divide both sides by 5

5y / 5  =  5 / 5

y  =  1

So, the solution to the system is

(x, y)  =  (2, 1)

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