Question

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`?

Answer 1

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 `(-6,-11)`

Explanation:

We solve the set of two equations as simultaneous equations.

Call this Equation 1: `y = 2x+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(y+5)+1 = 2y+10+1 =2y+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 `(-6, -11)`.

You could graph the lines to check this solution.

Answer 2

`(-6,-11)`

Explanation:

`"solve the equations using the method of "color(blue)"substitution"`

`color(red)(y)=2x+1to(1)`

`color(red)(y)=x-5to(2)`

`"since both equations express y in terms of x, we can"`
`"equate the right sides"`

`rArr2x+1=x-5`

`"subtract x from both sides"`

`2x-x+1=cancel(x)cancel(-x)-5`

`rArrx+1=-5`

`"subtract 1 from both sides"`

`xcancel(+1)cancel(-1)=-5-1`

`rArrx=-6`

`"substitute this value into either " (1)" or " (2)`

`"substituting in " (2)" and evaluating for y gives"`

`y=-6-5=-11`

`rArr" point of intersection "=(-6,-11)`
graph{(y-2x-1)(y-x+5)=0 [-40.54, 40.54, -20.28, 20.26]}