If the coefficients of the y-terms are equal and the coefficients of the x-terms are equal, the graphs of the two lines will be parallel.

Question

So a student has a claim that for any pair of linear relations, if the coefficients of the y-terms are equal and the coefficients of the x-terms are equal, the graphs of the two lines will be parallel.

For a pair of linear relations the slope has to be the same in both equations, so would an example of this claim being true be

y = 2x + 7 and y = 2x + - 7 ?

Then for an example of the claim being false, I know that the slope of the line must be different than the next to make the lines not parallel.

Y = 1x + 1 and Y = -1x + 1

Are the coefficients of both terms considered equal?

Answer 1

Hint : Two lines are parallel if and only if they have the same slope. If the line is given in the form $y=mx+b$, the slope is $m$.

Answer 2

To me, it is self-evident that two equations with the same slope will always be parallel. For example $y=2x+7$ is parallel to $y=2x+0.1$, which is parallel to $y=2x+b$, where $b$ is any positive or negative number. So perhaps you could expand a bit on what bothers you with this claim.

Answer 3

Suppose we have the system of equations $$\left\{\begin{eqnarray}ay&=bx+c\\ ay&=bx+c'. \end{eqnarray}\right.$$

Dividing by $a$, we obtain $$\left\{\begin{eqnarray}y&=\frac{b}{a}x+c\\ y&=\frac{b}{a}x+c', \end{eqnarray}\right.$$ which are parallel lines with slope $\frac{b}{a}$. In the second example you've given, you do not have the same value of $b$ for your coefficient on $x$ so the example fails.