# Question #f47ec

Aug 20, 2017

The key is to multiply one or both equations by values that will cause one variable to become zero.

$x = - 5 \mathmr{and} y = - 6$

#### Explanation:

In this example multiplying one of the equations by $- 1$ will result in one positive $x$ term and one negative $x$ term.
When added they will equal $0 x$

.Step $1.$ multiply the first equation by $- 1$

$- 1 \left(- x - y = 11\right) = + x + y = - 11$

Step $2$ add the two equations to get a $0$ value for the $x$ term.

$+ x + y = - 11$
$\underline{- x + 3 y = - 13} \text{ }$ which gives
$0 x + 4 y = - 24$

Step $3.$ solve for the other variable. $\left(y\right)$

$\frac{4 y}{4} = \frac{- 24}{4}$ so

$y = - 6$

step $4.$ substitute the value for one variable into one of the equations and solve for the other variable.

$- x - \left(- 6\right) = 11 \text{ }$ add $- 6$ to both sides

$- x + 6 - 6 = 11 - 6$ which gives

$- x = 5$ divide both sides by $- 1$

$\frac{- x}{-} 1 = \frac{5}{-} 1$ so

$x = - 5$

Aug 20, 2017

$x = - 5 , y = - 6$

#### Explanation:

$- x - y = 11.$

This means:

$- x = 11 + y$

Now we substitute $- x$ in the second equation:

$- x + 3 y = - 13$

We have:

$11 + y + 3 y = - 13$

$11 + 4 y = - 13$

$4 y = - 13 - 11$

$y = - \frac{24}{4} = - 6$

Now we substitute $y$ in either one of the two equations:

$- x - \left(- 6\right) = 11$

$- x + 6 = 11$

$- x = 11 - 6$

$x = - 5$

We can now substitute $x$ and $y$ in either equation to be sure that the answer is right.

Aug 20, 2017

$\left(- 5 , - 6\right)$

#### Explanation:

The goal here is to solve for one variable by first eliminating one variable, hence, the elimination method.

Given:

$- x - y = 11$

$- x + 3 y = - 13$

We can eliminate the $x$ variable since they have the same coefficient by subtracting both equations. Thus,

$\text{ } \cancel{- x} - y = 11$
$-$
$\text{ } \cancel{- x} + 3 y = - 13$

This yields:

$- 4 y = 24$

Now we solve for the variable $y$

$\frac{\cancel{- 4}}{\cancel{\textcolor{red}{- 4}}} y = \frac{24}{\textcolor{red}{- 4}}$

$y = - 6$

The next step is to find the value for the variable $x$ by substituting the value for $y$ into one of the original equations. I will use the first equation.

We substitute $- 6$ for $y$ in the first equation and solve for $x$

$- x - \left(\textcolor{red}{- 6}\right) = 11$

$- x + 6 = 11$

$- x + 6 \textcolor{red}{- 6} = 11 \textcolor{red}{- 6}$

$- x = 5$

$x = - 5$

We now have the solutions $x = - 5$ and $y = - 6$ or $\left(- 5 , - 6\right)$ but we must check if the solution checks out but plugging both values into both equations.

Equation 1:

$- \left(- 5\right) - \left(- 6\right) = 11$

$5 + 6 = 11$

$11 = 11$ TRUE

Equation 2:

$- \left(- 5\right) + 3 \left(- 6\right) = - 13$

$5 - 18 = 13$

$13 = 13$ TRUE

The values check out. The solution $\left(- 5 , - 6\right)$

Aug 20, 2017

$x = - 5 \mathmr{and} y = - 6$

#### Explanation:

The most important concept with the ELIMINATION method concerns ADDITIVE INVERSES.

Values which have the same number but opposite signs are called additive inverses. Their sum is $0$

$\left(+ 5\right) + \left(- 5\right) = 0 , \text{ } \left(- 12\right) + \left(+ 12\right) = 0$

To eliminate one of the variables, create additive inverses.
Other contributors have eliminated $x$, but let's look
at the $y$-terms instead

Notice that the $y$-terms already have opposite signs, but have different numbers. Create additive inverses.

$\textcolor{w h i t e}{\times \times} - x \textcolor{b l u e}{- y} = + 11 \text{ } A$
$\textcolor{w h i t e}{\times \times} - x \textcolor{b l u e}{+ 3 y} = - 13 \text{ } B$

$A \times 3 : - 3 x \textcolor{b l u e}{- 3 y} = + 33 \text{ "C" }$ now there are
$\textcolor{w h i t e}{\times x . \times} - x \textcolor{b l u e}{+ 3 y} = - 13 \text{ "B" }$ additive inverses

Add the equations together:

$C + B : \text{ "-4x " "= 20" } \leftarrow$ only $x$ values,
$\textcolor{w h i t e}{\times \times x . \times \times} x = - 5$

That that you know the value for $x$, substitute to find $y$

$\text{ } - x - y = 11$

$\text{ } - \left(- 5\right) - y = 11$

$5 - 11 = y$

$- 6 = y$