How do you solve the following system?: 4x -2y =3 , -3x -y = -6

Jun 20, 2018

See a solution process below:

Explanation:

Step 1) Solve the second equation for $y$:

$- 3 x - y = - 6$

$- 3 x - y + \textcolor{red}{y} + \textcolor{b l u e}{6} = - 6 + \textcolor{b l u e}{6} + \textcolor{red}{y}$

$- 3 x - 0 + \textcolor{b l u e}{6} = 0 + \textcolor{red}{y}$

$- 3 x + 6 = y$

$y = - 3 x + 6$

Step 2) Substitute $\left(- 3 x + 6\right)$ for $y$ in the first equation and solve for $x$:

$4 x - 2 y = 3$ becomes:

$4 x - 2 \left(- 3 x + 6\right) = 3$

$4 x - \left(2 \times - 3 x\right) - \left(2 \times 6\right) = 3$

$4 x - \left(- 6 x\right) - 12 = 3$

$4 x + 6 x - 12 = 3$

$\left(4 + 6\right) x - 12 = 3$

$10 x - 12 = 3$

$10 x - 12 + \textcolor{red}{12} = 3 + \textcolor{red}{12}$

$10 x - 0 = 15$

$10 x = 15$

$\frac{10 x}{\textcolor{red}{10}} = \frac{15}{\textcolor{red}{10}}$

$\frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{10}}} x}{\cancel{\textcolor{red}{10}}} = \frac{15}{10}$

$x = \frac{3}{2}$

Step 3) Substitute $\frac{3}{2}$ for $x$ in the solution to the second equation at the end of Step 1 and calculate $y$:

$y = - 3 x + 6$ becomes:

$y = \left(- 3 \times \frac{3}{2}\right) + 6$

$y = \frac{- 9}{2} + 6$

$y = \frac{- 9}{2} + \left(\frac{2}{2} \times 6\right)$

$y = \frac{- 9}{2} + \frac{12}{2}$

$y = \frac{- 9 + 12}{2}$

$y = \frac{3}{2}$

The Solution Is:

$x = \frac{3}{2}$ and $y = \frac{3}{2}$

Or

$\left(\frac{3}{2} , \frac{3}{2}\right)$

Jun 20, 2018

$y = \frac{3}{2} , x = \frac{3}{2}$

Explanation:

$4 x - 2 y = 3 - - - - - - - \left(1\right)$

$- 3 x - y = - 6 - - - - - - - \left(2\right)$

$\left(2\right) \times 2$

$\therefore - 6 x - 2 y = - 12 - - - - - - \left(3\right)$

$\left(1\right) - \left(3\right)$

$\therefore 10 x = 15$

$\therefore x = {\cancel{15}}^{3} / {\cancel{10}}^{2}$

$\therefore x = \frac{3}{2}$

$\text{substitute x"=3/2 "in} \left(2\right)$

$\therefore - 3 \left(\frac{3}{2}\right) - y = - 6$

$\therefore - \frac{9}{2} - y = - \frac{12}{2}$

$\therefore - y = - \frac{12}{2} + \frac{9}{2}$

$\therefore - y = - \frac{3}{2}$

$\therefore y = \frac{3}{2}$

~~~~~~~~~~~~~~~~
check:-

substitute $y = \frac{3}{2} \mathmr{and} x = \frac{3}{2} \text{in} \left(1\right)$

$\therefore 4 \left(\frac{3}{2}\right) - 2 \left(\frac{3}{2}\right) = 3$

$\therefore \frac{12}{2} - \frac{6}{2} = 3$

$\therefore {\cancel{6}}^{3} / {\cancel{2}}^{1} = 3$

$\therefore 3 = 3$