Question

solve the system of equations by the addition method 3x - 3y = 77x + 6y = 12

Answer 1

We will solve the following system of equations by the addition method.

$\mleft\{\begin{aligned}3x-3y=7 \\ 7x+6y=12\end{aligned}\mright.$

For doing so, we multiply the first and the second equation by some numbers, with the intention of making the coefficients of the variable x opposites.

Having this in mind, we will multiply the first equation for -7, using the distributive property.

$\begin{gathered} -7(3x-3y=7) \\ -21x+21y=-49 \end{gathered}$

And the second equation by 3.

$\begin{gathered} 3(7x+6y=12) \\ 21x+18y=36 \end{gathered}$

And we obtain,

$\mleft\{\begin{aligned}-21x+21y=-49 \\ 21x+18y=36\end{aligned}\mright.$

The next step is to sum both equations:

$\begin{gathered} -21x+21x+21y+18y=-49+36 \\ 0+21y+18y=-13 \\ 39y=-13 \\ y=-\frac{13}{39}=-\frac{1}{3} \end{gathered}$

The last step is to replace the value on any of the two equations. We would do it on the first equation:

$\begin{gathered} 3x-3(-\frac{1}{3})=7 \\ 3x+1=7 \\ 3x=7-1 \\ 3x=6 \\ x=\frac{6}{3}=2 \end{gathered}$

This means that the solution is:

$(x,y)=(2,-\frac{1}{3})$