# How do you solve 9x+6y=3 and -3x+y=14 using substitution?

Jun 1, 2018

First, you take 1 equation and you make it so that the $y$ is the main thing like $y =$something... Then you take that and you substitute it into your other equation to get $x =$something...You then take that and you substitute it into your first equation and done.

#### Explanation:

NOTE:
color color(gold)(gold and the bolded text is the main stuff
Other writing is explanations

So, to start you got these 2 equations...
color(gold)(9x + 6y = 3
color(gold)(-3x + y = 14

Let's call equation 1 A and equation 2 B. So...
$9 x + 6 y = 3$: color(red)(A
$- 3 x + y = 14$: color(blue)(B

So to start substituting, you first take color(blue)(B because it has the lowest number of '$y$'s...
$- 3 x + y = 14$ color(blue)(B
$9 x + 6 y = 3$ color(red)(A
Equation color(blue)(B has the lowest number of $y$.

Now, you take equation color(blue)(B and you twist and turn it to make the $y$ in the equation the main letter. Like so...

color(gold)(-3x + y = 14 | This is your starting equation
$y = 14 + 3 x$ | you move over the '$- 3 x$' to the other side of the '$=$' sign. Since it was a 'negative' number before, going over to the other side makes '$- 3 x$' a positive number, '$3 x$'.
color(gold)(y = 14 + 3x | now '$y$' is the main letter.

Now you have this, you get your other equation (equation color(red)(A) and you 'combine' it with your new equation color(blue)(B. Like so...

$9 x + 6 y = 3$: color(red)(A
$y = 14 + 3 x$: NEW color(blue)(B

You substitute your NEW color(blue)(B into color(red)(A...
$9 x + 6 \textcolor{g o l d}{\left(14 + 3 x\right)} = 3$ | In the NEW color(blue)(B, it says the $y =$ 'that'. Therefore, you can make the $y$ in color(red)(A ($6 y$) also equal 'that'.

Now you just have to do some switch-a-roos and stuff to get your answers: x = something and y = something, like this...

color(gold)(9x + 6(14 + 3x) = 3 | what you got now
$9 x + 84 + 18 x = 3$ | expanded the brackets
$27 x + 84 = 3$ | added the similar numbers
$27 x = 3 - 84$ | moved $84$ to the other side
$27 x = - 81$
$x = - \frac{81}{27}$ | moved $27$ to the other side.
color(gold)(x = -3 | your first answer.

Now...
color(gold)(-3x + y = 14: color(blue)(B | The old equation color(blue)(B
Since $x = - 3$ so...
$- 3 \left(- 3\right) + y = 14$ | substituting $x = - 3$
$9 + y = 14$
$y = 14 - 9$ | moving 9 to the other side
color(gold)(y = 5 | your second answer.

That's how you do it.