# How do you solve the following linear system: -9x - 5y = -1, 3x + y = 1 ?

Sep 19, 2016

There are two methods: Substitution and addition, subtraction.
For this problem addition or subtraction would be the preferred method.
$x = \frac{2}{3} \mathmr{and} y = - 1$

#### Explanation:

Multiplying the second equation by 3 makes adding the two equations easy

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

$9 x + 3 y = + 3$

$- 9 x - 5 y = - 1$
$\underline{+ 9 x + 3 y = + 3}$
$\text{ } 0 x - 2 y = + 2$ This is what it gives

Now solve for y by dividing both sides by $- 2$

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

This gives $y = - 1$

Put $- 1$ into either equation for $y$ and solve for $x$

$3 x - 1 = + 1 \text{ }$ Add + 1 to both sides of the equation

$3 x - 1 + 1 = + 1 + 1$

This gives

$3 x = 2 \text{ }$ Divide both sides by 3

$\frac{3 x}{3} = \frac{+ 2}{3} \text{ }$ This gives

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

The point of intersection is $\left(\frac{2}{3} , - 1\right)$

The substitution method would also work but in this case would be more difficult. An alternative would be to graph both equations and then find the point of intersection.