# What are x and y if y = 4x + 3 and 2x + 3y = -5?

May 15, 2018

$x = - 1$ and $y = - 1$

#### Explanation:

show below

$y = 4 x + 3$..........1

$2 x + 3 y = - 5$..........2

put 1 in 2

$2 x + 3 \left(4 x + 3\right) = - 5$

$2 x + 12 x + 9 = - 5$

$14 x = - 14$

$x = - 1$

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

May 15, 2018

Through substitution or elimination, we can determined that $x = - 1$ and $y = - 1$.

#### Explanation:

There are two ways to algebraically solve for $x$ and $y$.

Method 1: Substitution

Through this method, we solve to a variable in one equation and plug it in to the other. In this case, we already know the value of $y$ in the first equation. Therefore, we can substitute it it for $y$ in the second equation and solve for $x$.

$y = 4 x + 3$
$2 x + 3 \left(4 x + 3\right) = - 5$
$2 x + 12 x + 9 = - 5$
$14 x = - 14$
$x = - 1$

Now, we just need to plug $x$ back in to one of the equations to solve for $y$. We can use the first equation because $y$ is already isolated, but both will yield the same answer.

y=4(-1)+3)
$y = - 4 + 3$
$y = - 1$

Therefore, $x$ is $- 1$ and $y$ is $- 1$.

Method 2: Elimination

Through this method, the equations are subtracted so that one of the variables is eliminated. To do this, we must isolate the constant number. In other words, we put $x$ and $y$ on the same side, like in the second equation.

$y = 4 x + 3$
$0 = 4 x - y + 3$
$- 3 = 4 x - y$

Now, the equations are both in the same form. However, to eliminate one of the variables, we must get $0$ when the equations are subtracted. This means we must have the same coefficients on the variable. For this example, let's solve for $x$. In the first equation, $x$ has a coefficient of $4$. Thus, we need $x$ in the second equation to have the same coefficient. Because $4$ is $2$ times its current coefficient of $2$, we need to multiply the entire equation by $2$ so it stays equivalent.

$2 \left(2 x + 3 y\right) = 2 \left(- 5\right)$
$4 x + 6 y = - 10$

Next, we can subtract the two equations.

$4 x + 6 y = - 10$
$- \left(4 x - y = - 3\right)$
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$0 x + 7 y = - 7$

$7 y = - 7$
$y = - 1$

As with the first method, we plug this value back in to find $x$.

$- 1 = 4 x + 3$
$- 4 = 4 x$
$- 1 = x$

Therefore, $x$ is $- 1$ and $y$ is $- 1$.