a) If x−2y=−8, and 3x−6y=−12, what are the values of x and y? b) How do you prove your answer from (a) graphically?
1 Answer
a). The first step in solving by substitution is always solving for one of the variables. Since x in the second equation has coefficient
x−2y=−8→x=2y−8
We now substitute this into the first equation.
3(2y−8)−6y=−12
6y−24−6y=−12
0y=12
This is true for no real value of
b). Let's do a little bit of work with the first equation.
3x−6y=−12
We factor out a
3(x−2y)=−12
Divide both sides by
x−2y=−4
We get an equation that is identical to the second on the left-hand side, but different on the right-hand side. What does this mean?
Suppose we were to graph both lines. We would first convert to slope-intercept form.
x−2y=−4→−2y=−4−x→y=12x+2
For the second equation:
x−2y=−8→−2y=−8−x→y=12x+4
These lines have equal slopes but different y-intercepts. This means that these are parallel lines, which is graphical proof that they will never intersect.
Hopefully this helps!
