**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.

**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

Hence, the solution to the system is

(x, y) = (2, 1)

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