How do you solve the system 5x + 6y = 2  and -2x + 3y = 37 ?

Mar 18, 2018

See a solution process below:

Explanation:

Step 1) Solve each equation for $6 y$:

• Equation 1:

$5 x + 6 y = 2$

$5 x - \textcolor{red}{5 x} + 6 y = 2 - \textcolor{red}{5 x}$

$0 + 6 y = 2 - 5 x$

$6 y = 2 - 5 x$

• Equation 2:

$- 2 x + 3 y = 37$

$- 2 x + \textcolor{red}{2 x} + 3 y = 37 + \textcolor{red}{2 x}$

$0 + 3 y = 37 + 2 x$

$3 y = 37 + 2 x$

$\textcolor{red}{2} \times 3 y = \textcolor{red}{2} \left(37 + 2 x\right)$

$6 y = \left(\textcolor{red}{2} \times 37\right) + \left(\textcolor{red}{2} \times 2 x\right)$

$6 y = 74 + 4 x$

Step 2) Because the left side of both equations are the same we can equate the right side of both equations and solve for $x$:

$2 - 5 x = 74 + 4 x$

$2 - \textcolor{red}{2} - 5 x - \textcolor{b l u e}{4 x} = 74 - \textcolor{red}{2} + 4 x - \textcolor{b l u e}{4 x}$

$0 + \left(- 5 - \textcolor{b l u e}{4}\right) x = 72 + 0$

$- 9 x = 72$

$\frac{- 9 x}{\textcolor{red}{- 9}} = \frac{72}{\textcolor{red}{- 9}}$

$\frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{- 9}}} x}{\cancel{\textcolor{red}{- 9}}} = - 8$

$x = - 8$

Step 3) Substitute $- 8$ into either of the equations in Step 1 and solve for $y$:

$3 y = 37 + 2 x$ becomes:

$3 y = 37 + \left(- 8 \times 2\right)$

$3 y = 37 - 16$

$3 y = 21$

$\frac{3 y}{\textcolor{red}{3}} = \frac{21}{\textcolor{red}{3}}$

$\frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{3}}} y}{\cancel{\textcolor{red}{3}}} = 7$

$y = 7$

The Solution Is:

$x = - 8$ and $y = 7$

Or

$\left(- 8 , 7\right)$