# Solve the given system of equations by the addition method. 6x-5y=12 ; x+y=3

## Question:

Solve the given system of equations by the addition method.

{eq}6x - 5y = 12 {/eq} ; {eq}x + y = 3 {/eq}

In the addition method, we solve a system of two equations of two variables by adding the equations. By doing so, we get a linear equation in one variable, which we can solve easily.

The given equations are:

\begin{align} 6x-5y&=12& \rightarrow (1)\\[0.4cm] x+y&=3& \rightarrow (2) \end{align}

Multiplying the equation (2) by {eq}-6 {/eq} on both sides,

$$-6(x+y)= -6(3) \Rightarrow -6x-6y =-18 \,\,\,\,\,\, \rightarrow (3)$$

$$(6x-5y) +(-6x-6y) = 12+(-18) \\[0.4cm] 6x-6x-5y-6y=-6 \\[0.4cm] \text{Combining the like terms}, \\[0.4cm] -11y =-6\\[0.4cm] \text{Dividing both sides by -11}, \\[0.4cm] y= \dfrac{6}{11}$$

Substituting this in (2),

$$x+\dfrac{6}{11} = 3 \\[0.4cm] \text{Subtracting }\dfrac{6}{11} \text{ from both sides}, \\[0.4cm] x = 3 - \dfrac{6}{11} = \dfrac{27}{11}$$

Therefore, the solution of the given system is, {eq}\boxed{\mathbf{x=\frac{27}{11} \text { and } y=\frac{6}{11}}} {/eq}. 