# How do you write the the ordered pair that is the solution to the following system of equations: y= 2x + 1 and y = x -5?

May 21, 2017

The ordered pair will simply be the point at which these two lines cross. We solve the equations as simultaneous equations.

The solution is the point $\left(- 6 , - 11\right)$

#### Explanation:

We solve the set of two equations as simultaneous equations.

Call this Equation 1: $y = 2 x + 1$
Call this Equation 2: $y = x - 5$

Rearrange Equation 2 to make $x$ the subject:

$x = y + 5$

Substitute this value of $x$ into Equation 1:

$y = 2 \left(y + 5\right) + 1 = 2 y + 10 + 1 = 2 y + 11$

Rearranging:

$- y = 11$

$y = - 11$

We can substitute this into either equation to find the value of $x$:

$x = - 6$

That means the intersection of the lines is at the point $\left(- 6 , - 11\right)$.

You could graph the lines to check this solution.

May 21, 2017

$\left(- 6 , - 11\right)$

#### Explanation:

$\text{solve the equations using the method of "color(blue)"substitution}$

$\textcolor{red}{y} = 2 x + 1 \to \left(1\right)$

$\textcolor{red}{y} = x - 5 \to \left(2\right)$

$\text{since both equations express y in terms of x, we can}$
$\text{equate the right sides}$

$\Rightarrow 2 x + 1 = x - 5$

$\text{subtract x from both sides}$

$2 x - x + 1 = \cancel{x} \cancel{- x} - 5$

$\Rightarrow x + 1 = - 5$

$\text{subtract 1 from both sides}$

$x \cancel{+ 1} \cancel{- 1} = - 5 - 1$

$\Rightarrow x = - 6$

$\text{substitute this value into either " (1)" or } \left(2\right)$

$\text{substituting in " (2)" and evaluating for y gives}$

$y = - 6 - 5 = - 11$

$\Rightarrow \text{ point of intersection } = \left(- 6 , - 11\right)$
graph{(y-2x-1)(y-x+5)=0 [-40.54, 40.54, -20.28, 20.26]}