Consider the equation $ax + by = c$ in 2-space and the slope for that equation where $a$ and $b$ are real numbers.

Question

Consider the equation $ax + by = c$ in 2-space and the slope for that equation where a and b are real numbers.

Explain using multiple representations why it is equivalent to say that a linear equation moves vertically $−a/b$ units for every $1$ unit of movement horizontally as it is to say that linear equations move vertically $-a$ units for every b units of horizontal movement.

I am having a hard time showing this. Should I plug in values and show rise/run graphically and maybe use a table? How do I explain this?

Answer 1

The very definition of slope is rise over run, or $\frac {\Delta y}{\Delta x}$. We know from this problem that the slope of the line is $-\frac {a}{b}$. This means that for every $b$ units that $x$ has moved, $y$ moves $a$ units. Because this is a linear equation, we can see that for every $1$ unit that $x$ moves, $y$ moves $\frac {-a}{b}$. Try it for yourself! You can plot lines and see how much it rises/runs and compare it to $-\frac {a}{b}$.

Answer 2

You’ve taken the first step by solving the equation for $y$ to get $$y = -\frac ab x + c.$$ This requires that $b\ne0$, so you might have to examine that case separately. Now you need to work out the effect of moving one unit horizontally. Let’s say that you’re currently at $x=x_0$ with corresponding $y$-coordinate $y_0 = -\frac abx_0+c$. If you move one unit to the right, the $x$-coordinate becomes $x_1 = x_0+1$, so plug that into the equation to get: $$y_1 = -\frac ab(x_0+1)+c.$$ You want to see how much $y$ changed because of this unit step, so write down an expression for $y_1-y_0$ in terms of $x_0$ and simplify. If you do this correctly, $x_0$ will be eliminated, which means that the change in $y$ due to a unit step doesn’t depend on where you start—it’s the same everywhere.