**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.

**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 = -17

2x + 15y = 7

**Answer :**

**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

Add 12 to both sides.

(-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)

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