# Solve the system of equations. 4x-5y=38 ; 7x-8y=-22

## Question:

Solve the system of equations.

{eq}4x - 5y = 38 {/eq} ; {eq}7x - 8y = -22 {/eq}

## Elimination Method:

(i) The elimination method is used to solve a system of two equations in two variables.

(ii) In this method, the given equations are added or subtracted to eliminate one variable.

(iii) The resultant equation in one variable can be solved using algebraic operations.

The given two equations are:

\begin{align} 4x-5y&=38 & \rightarrow (1) \\[0.4cm] 7x-8y&=-22& \rightarrow (2) \end{align}

Multiply both sides of (1) by {eq}-7 {/eq}:

$$-7(4x-5y=38 ) \Rightarrow -28x+35y = -266 \,\,\,\,\,\,\,\,\,\rightarrow (3)$$

Multiply both sides of (2) by {eq}4 {/eq}:

$$4(7x-8y=-22) \Rightarrow 28x -32y= -88 \,\,\,\,\,\,\,\,\,\rightarrow (4)$$

$$( -28x+35y) +(28x -32y) = -266 -88 \\[0.4cm] 3y = -354 \\[0.4cm] \text{Dividing both sides by 3}, \\[0.4cm] y=-118$$

Substitute this in (1):

$$4x-5(-118) = 38 \\[0.4cm] 4x+590 =38 \\[0.4cm] \text{Subtracting 590 from both sides}, \\[0.4cm] 4x = -552 \\[0.4cm] \text{Dividing both sides by 4}, \\[0.4cm] x=-138$$

Therefore, the solution of the given system is, {eq}\boxed{\mathbf{x=-138 \text { and }y=-118}} {/eq}.