How do you solve y=4x-9 and y=x-3 using substitution?

Mar 24, 2017

See the entire solution process below:

Explanation:

Step 1) Because the first equation is already solve for $y$, substitute $4 x - 9$ for $y$ in the second equation and solve for $x$:

$y = x - 3$ becomes:

$4 x - 9 = x - 3$

$4 x - 9 + \textcolor{red}{9} - \textcolor{b l u e}{x} = x - 3 + \textcolor{red}{9} - \textcolor{b l u e}{x}$

$4 x - \textcolor{b l u e}{1 x} - 9 + \textcolor{red}{9} = x - \textcolor{b l u e}{x} - 3 + \textcolor{red}{9}$

$\left(4 - \textcolor{b l u e}{1}\right) x - 0 = 0 + 6$

$3 x = 6$

$\frac{3 x}{\textcolor{red}{3}} = \frac{6}{\textcolor{red}{3}}$

$\frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{3}}} x}{\cancel{\textcolor{red}{3}}} = 2$

$x = 2$

Step 2) Substitute $2$ for $x$ in the first equation and calculate $y$:

$y = 4 x - 9$ becomes:

$y = \left(4 \times 2\right) - 9$

$y = 8 - 9$

$x = 2$

The solution is: $x = 2$ and $x = 2$ or $\left(2 , - 1\right)$

Mar 24, 2017

$\left(2 , - 1\right)$

Explanation:

Since both equations are $y =$, substitute the first equation for the second "y":

$4 x - 9 = x - 3$

Now solve for $x$:
$4 x - x - 9 = x - x - 3$

$3 x - 9 = - 3$

$3 x - 9 + 9 = - 3 + 9$

$3 x = 6$

$\frac{3}{3} x = \frac{6}{3} = 2$

Now substitute $x$ into either equation to find $y$:

$y = 2 - 3 = - 1$

Check by substituting both putting both values into the first equation:
$- 1 = 4 \left(2\right) - 9$
$- 1 = 8 - 9$
$- 1 = - 1$ TRUE

So $x = 2 , y = - 1$

Which is the point of intersection: $\left(2 , - 1\right)$

Mar 24, 2017

$\left(2 , - 1\right)$

Explanation:

Labelling the equations.

$\textcolor{red}{y} = 4 x - 9 \to \left(1\right)$

$\textcolor{red}{y} = x - 3 \to \left(2\right)$

Since both equations have y as the subject we can equate the right sides.

$\Rightarrow 4 x - 9 = x - 3$

subtract x from both sides.

$4 x - x - 9 = \cancel{x} \cancel{- x} - 3$

$\Rightarrow 3 x - 9 = - 3$

add 9 to both sides.

$3 x \cancel{- 9} \cancel{+ 9} = - 3 + 9$

$\Rightarrow 3 x = 6$

divide both sides by 3

$\frac{\cancel{3} x}{\cancel{3}} = \frac{6}{3}$

$\Rightarrow x = 2$

Substitute this value into either of the equations

$\text{Substitute " x=2" in } \left(2\right)$

$\Rightarrow y = 2 - 3 = - 1$

$\textcolor{b l u e}{\text{As a check}}$

$\text{Substitute " x=2" in } \left(1\right)$

$\Rightarrow y = \left(4 \times 2\right) - 9 = 8 - 9 = - 1 \to \text{ true}$

$\Rightarrow \left(2. - 1\right) \text{ is the point of intersection}$
graph{(y-4x+9)(y-x+3)=0 [-10, 10, -5, 5]}