# How do you solve the system of equations -2x - 2y = - 4 and 2x + 3y = 9?

Oct 14, 2017

Determine the point of intersection by using substitution and solving for the variables.

#### Explanation:

Solving a system of equations imply we find the point of interception.

To do this, we must use substitution to find one variable, and use that value to determine the exact value of the other.

Here, we'll isolate $x$.

$- 2 x - 2 y = - 4$

$- 2 x = 2 y - 4$

$x = - y + 2$

Now that we have our variable expression, we can sub this into the other equation and solve for $y$.

$2 x + 3 y = 9$

$2 \left(- y + 2\right) + 3 y = 9$

$- 2 y + 4 + 3 y = 9$

$4 + y = 9$

$y = 5$

Now that we know our $y$ value, we can sub it into any equation to solve for $x$.

$x = - y + 2$

$x = - \left(5\right) + 2$

$x = - 5 + 2$

x=~3

Thus, our point of intersection is (~3, 5).

If we graph the two equations, we can see the point of intersection is in fact (~3, 5).

graph{(-2x-2y+4)(2x+3y-9)=0 [-8.875, 11.125, -0.36, 9.64]}

Hope this helps :)

Oct 14, 2017

Refer to the explanation.

#### Explanation:

Solve system:

Equation 1: $- 2 x - 2 y = - 4$

Equation 2: $2 x + 3 y = 9$

Both equations are linear equations in standard form: $\text{Ax+By=C}$.

The resulting point $\left(x , y\right)$ is the point at which the two lines intersect on a graph.

The system of equations will be solved using substitution.

Solve Equation 1 for $x$.

$- 2 x - 2 y = - 4$

Add $2 y$ to both sides.

$- 2 x = - 4 + 2 y$

Divide both sides by $- 2$.

$x = 2 - y$

Solve Equation 2 for $y$.

Substitute $2 - y$ for $x$ in Equation 2 and solve for $y$.

$2 x + 3 y = 9$

$2 \left(2 - y\right) + 3 y = 9$

Simplify.

$4 - 2 y + 3 y = 9$

Subtract $4$ from both sides.

$- 2 y + 3 y = 9 - 4$

Simplify.

$y = 5$

Substitute $5$ for $y$ in Equation 1 and solve for $x$.

$- 2 x - 2 y = - 4$

$- 2 x - 2 \left(5\right) = - 4$

Simplify.

$- 2 x - 10 = - 4$

Add $10$ to both sides.

$- 2 x = - 4 + 10$

Simplify.

$- 2 x = 6$

Divide both sides by $- 2$.

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

Simplify.

$x = - 3$

Point of Intersection: $\left(- 3 , 5\right)$