# How do you solve the system 3x-y=4 and 2x-3y=-9 using substitution?

Feb 13, 2017

See the entire solution process below:

#### Explanation:

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

$3 x - y = 4$

$- \textcolor{red}{3 x} + 3 x - y = - \textcolor{red}{3 x} + 4$

$0 - y = - 3 x + 4$

$- y = - 3 x + 4$

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

$y = 3 x - 4$

Step 2) Substitute $3 x - 4$ for $y$ in the second equation and solve for $x$:

$2 x - 3 y = - 9$ becomes:

$2 x - 3 \left(3 x - 4\right) = - 9$

$2 x - 9 x + 12 = - 9$

$- 7 x + 12 - \textcolor{red}{12} = - 9 - \textcolor{red}{12}$

$- 7 x + 0 = - 21$

$- 7 x = - 21$

$\frac{- 7 x}{\textcolor{red}{- 7}} = - \frac{21}{\textcolor{red}{- 7}}$

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

$x = 3$

Step 3) Substitute $3$ for $x$ in the solution to the first equation at the end of Step 1 and calculate $y$:

$y = 3 x - 4$ becomes:

$y = \left(3 \times 3\right) - 4$

$y = 9 - 4$

$y = 5$

The solution is: $x = 3$ and $y = 5$ or $\left(3 , 5\right)$

Feb 13, 2017

$\textcolor{red}{\text{Extreme detail}}$ given for determining $x$ using first principles.

The shared point of these two equations is$\text{ } \left(x , y\right) \to \left(3 , 5\right)$

#### Explanation:

Given:
$3 x - y = 4 \text{ } \ldots \ldots \ldots \ldots . . E q u a t i o n \left(1\right)$
$2 x - 3 y = - 9 \text{ } \ldots \ldots \ldots E q u a t i o n \left(2\right)$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Determine the value of x}}$

Consider $E q u a t i o n \left(1\right)$

Add $\textcolor{red}{y}$ to both sides

" "color(green)(3x-ycolor(red)(+y)" "=" "4 color(red)(+y))

But $- y + y = 0$ giving:

$\text{ } 3 x + 0 = 4 + y$

$\text{ } 3 x = 4 + y$

Subtract $\textcolor{red}{4}$ from both sides

" "color(green)(3xcolor(red)(-4)" "=" "4color(red)(-4)+y)

$\text{ } 3 x - 4 = y$

$\text{ "y=3x-4" } \ldots \ldots E q u a t i o n \left({1}_{a}\right)$

As we have just used equation(1) we now need to use equation(2)

Using $E q u a t i o n \left({1}_{a}\right)$ substitute for $y$ in $E q u a t i o n \left(2\right)$

$\textcolor{g r e e n}{2 x - 3 \textcolor{red}{y} = - 9 \text{ "->" } 2 x - 3 \left(\textcolor{red}{3 x - 4}\right) = - 9}$

$\text{ } 2 x - 9 x + 12 = - 9$

$\text{ } - 7 x + 12 = - 9$

Subtract 12 from both sides

" "color(green)(-7x+12color(red)(-12)" "=" "-9color(red)(-12))

$\text{ } - 7 x + 0 = - 21$

Divide both sides by -7

$\text{ } \textcolor{g r e e n}{\frac{- 7}{\textcolor{red}{- 7}} \textcolor{w h i t e}{.} x = \frac{- 21}{\textcolor{red}{- 7}}}$

$\text{ } + 1 \times x = + 3$

$\text{ } x = 3$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Determine the value of y}}$

I chose equation 1 as it is the most strait forward one to use.

Substitute for x in equation 1 giving:

$3 x - y = 4 \text{ "->" } 3 \left(3\right) - y = 4$

$y = 5$