Example 4.9.15 With Slope Triangles

We can always rearrange \(3x+4y=12\) into slope-intercept form (4.5.1), and then graph it with the slope triangle method:

\begin{align*} 3x+4y\amp=12\\ 4y\amp=12\subtractright{3x}\\ 4y\amp=-3x+12\\ y\amp=\divideunder{-3x+12}{4}\\ y\amp=-\frac{3}{4}x+3 \end{align*}

With the \(y\)-intercept at \((0,3)\) and slope \(-\frac{3}{4}\text{,}\) we can graph the line using slope triangles:

This is a graph of the line 3x+4y=12. The following points on the line are plotted: (-4,6),(0,3),(4,0). There  are  two slope triangles. One starts at (-4,6), passes (0,6), and ends at (0,3). One starts at (0,3), passes (4,3), and  ends at   (4,0).
Figure 4.9.16 Graphing \(3x+4y=12\) using slope triangles

Compared with the intercepts method, the slope triangle method takes more time, but shows more points with slope triangles, and thus a more accurate graph. Also sometimes (as with Example 4.7.14) when we graph a standard form equation like \(2x-3y=0\text{,}\) the intercepts method doesn't work because both intercepts are actually at the same point, and we have to resort to something else like slope triangles anyway.