Micah walks on a treadmill at 4 miles per hour. He has walked 2 miles when Luke starts running at 6 miles per hour on the treadmill next to him. If their rates continue, will Luke's distance ever equal Micah's distance? Explain.

If [tex]x=[/tex] The number of hours and [tex]y=[/tex] The number of miles, then the system of equations is [tex]\left \{ {{y=4x+2} \atop {y=6x}} \right.[/tex]where Micah is the first equation and Luke is the second.

Solve by Substitution:

1) Solve [tex]y=4x+2[/tex] for [tex]y[/tex]:

[tex]y=4x+2[/tex]

2) Substitute [tex]4x+2[/tex] for [tex]y[/tex] in [tex]y=6x[/tex]:

[tex]y=6x[/tex]

[tex]4x+2=6x[/tex]

3) Add -6x to both sides:

[tex]4x+2+-6x=6x+-6x[/tex]

4) Simplify both sides of the equation:

[tex]-2x+2=0[/tex]

5) Add -2 to both sides:

[tex]-2x+2+-2=0+-2[/tex]

6) Simplify both sides of the equation:

[tex]-2x=-2[/tex]

7) Divide both sides by -2:

[tex]\frac{-2x}{-2} =\frac{-2}{-2}[/tex]

8) Simplify both sides of the equation:

[tex]x=1[/tex]

9) Substitute [tex]1[/tex] for [tex]x[/tex] in [tex]y=4x+2[/tex]:

[tex]y=4x+2[/tex]

[tex]y=(4)(1)+2[/tex]

10) Simplify both sides of the equation:

[tex]y=6[/tex]

Therefore, They meet at the 1st hour at the 6th mile

Solve by elimination:

You can't solve the system of equations by elimination

Solve by graphing:

They cross at (1, 6), which supports my answer.