How do you solve the following linear system:  -x+3y=-9 , 5x-2y=-35 ?

Nov 15, 2015

$x = \frac{29}{4}$ I have shown in detail how to obtain the value of $x$ but will let you solve for $y$. This can be done by substituting $\frac{29}{4} \text{ for } x$

Explanation:

Given:
$- x + 3 y = - 9. \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . \left(1\right)$
$5 x - 2 y = 35. \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \left(2\right)$

$\textcolor{red}{\text{~~~~~~~~~~~~~~~~~~ All calculation shown ~~~~~~~~~~~~~~~}}$

For equation (1): making $y$ the dependant variable (y=..)

Add $\textcolor{b l u e}{x}$ to both sides so that it is removed from the left.

$\left(3 y - x\right) \textcolor{b l u e}{+ x} = \left(- 9\right) \textcolor{b l u e}{+ x}$

$\textcolor{b r o w n}{\text{The brackets serve no purpose other than to show what}}$
$\textcolor{b r o w n}{\text{is being altered or to group things so that they are obvious.}}$

$3 y = x - 9$

Divide both sides by 3

$\frac{3}{\textcolor{b l u e}{3}} \times y = \frac{x}{\textcolor{b l u e}{3}} - \frac{9}{\textcolor{b l u e}{3}}$

$\textcolor{g r e e n}{y = \frac{1}{3} x - 3. \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . \left({1}_{a}\right)}$

$\textcolor{red}{\text{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}}$

For equation (2): making $y$ the dependant variable (y=..)

By sight!

$\textcolor{g r e e n}{y = \frac{5}{2} x - \frac{35}{2.} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . . \left({2}_{a}\right)}$

$\textcolor{red}{\text{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}}$

Both equation $\left({1}_{a}\right)$ and $\left({2}_{a}\right)$ have a common value in y so
adopting $\text{Equation "(1_a) =y = "Equation } \left({2}_{a}\right)$ to solve for $x$

$\textcolor{b l u e}{\frac{1}{2} x - 3} \textcolor{b r o w n}{= y =} \textcolor{b l u e}{\frac{5}{2} x - \frac{35}{2}}$

Giving:

$\textcolor{b l u e}{\frac{1}{2} x - 3} \textcolor{b r o w n}{=} \textcolor{b l u e}{\frac{5}{2} x - \frac{35}{2}}$

Collecting like terms:

$\frac{5}{2} x - \frac{1}{2} x = \frac{35}{2} - 3$

$2 x = \frac{29}{2}$

Divide both sides by 2 giving

$\left(2 x\right) \div i \mathrm{de} 2 = \left(\frac{29}{2}\right) \div i \mathrm{de} 2$

$\frac{2}{2} x = \frac{29}{2} \times \frac{1}{2}$

$\textcolor{g r e e n}{x = \frac{29}{4.} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . \left(3\right)}$

$\textcolor{red}{\text{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}}$

Now substitute for $x$ using equation (3) into either equation $\left({1}_{a}\right) \text{ or } \left({2}_{a}\right)$ to determine the value of y.

I will let you do that!