Solve the following system by the addition method. 5x + 3y = -9, 5x - 7y = 9

Question:

Solve the following system by the addition method.

{eq}5x + 3y = -9 {/eq}

{eq}5x - 7y = 9 {/eq}

Addition Method:

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.

Answer and Explanation:

The given equations are:

$$\begin{align} 5x + 3y& = -9& \rightarrow (1)\\ 5x - 7y &= 9& \rightarrow (2) \end{align} $$

Multiplying the equation (1) by {eq}-1 {/eq} on both sides,

$$-1(5x + 3y) = -1(-9) \Rightarrow -5x-3y =9 \,\,\,\,\,\, \rightarrow (3) $$

Adding (2) and (3):

$$(5x-7y) +(-5x-3y) = 9+9 \\[0.4cm] 5x-5x-7y-3y=18 \\[0.4cm] \text{Combining the like terms}, \\[0.4cm] -10y =18\\[0.4cm] \text{Dividing both sides by -10}, \\[0.4cm] y= \dfrac{18}{-10}= \dfrac{-9}{5} $$

Substituting this in (1),

$$5x + 3 \left( \dfrac{-9}{5}\right) = -9 \\[0.4cm] 5x - \dfrac{27}{5}=-9 \\[0.4cm] \text{Adding } \dfrac{27}{5} \text{ on both sides}, \\[0.4cm] 5x = -9 + \dfrac{27}{5} = \dfrac{-45}{5}+ \dfrac{27}{5} = \dfrac{-18}{5} \\[0.4cm] \text{Dividing both sides by 5}, \\[0.4cm] x= \dfrac{-18}{25} $$

Therefore, the solution of the given system is, {eq}\boxed{\mathbf{x=-\frac{18}{25} \text { and } y=-\frac{9}{5}}} {/eq}.


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