Slope–Intercept $(y = mx + b)$
Algebra, Lines
Intro: Convert to y = mx + b either from two points or directly from a linear equation.
Worked example
- From points $(−2,5)$ and $(4,−1)$, find $y=mx+b$.
- Given points: $(x_1,y_1)=(-2,5)$ and $(x_2,y_2)=(4,-1)$.
- Slope formula: $$m=\frac{y_2-y_1}{x_2-x_1}.$$
- Compute numerator: $y_2-y_1=-1-5=-6$.
- Compute denominator: $x_2-x_1=4-(-2)=6$.
- Therefore slope: $$m=\frac{-6}{6}=-1.$$
- Use point–slope form with $(-2,5)$: $$y-y_1=m(x-x_1)\;\Rightarrow\; y-5=-1\,(x-(-2))= -1\,(x+2).$$
- Distribute and simplify: $$y-5=-x-2 \;\Rightarrow\; y=-x-2+5=-x+3.$$
- Thus slope–intercept form is $$\boxed{y=-x+3}.$$
- Read off parameters: $$\boxed{m=-1},\quad \boxed{b=3}.$$
- Check with the second point $(4,-1)$: substitute $x=4$ into $y=-x+3$ to get $y=-4+3=-1$ ✔️.
- Rewrite $3x-2y=8$ as $y=mx+b$.
- Start from the standard form: $$3x-2y=8.$$
- Isolate the $y$-term by subtracting $3x$ from both sides: $$-2y=-3x+8.$$
- Divide both sides by $-2$: $$y=\frac{-3x+8}{-2}=\frac{3}{2}x-4.$$
- Hence the slope–intercept form is $$\boxed{y=\tfrac{3}{2}x-4}.$$
- Read off parameters: $$\boxed{m=\tfrac{3}{2}},\quad \boxed{b=-4}.$$
- Quick check: put $x=0$ into the original equation $3x-2y=8$ → $-2y=8$ → $y=-4$, which matches the intercept $b=-4$ ✔️.
FAQs
Can I use decimals?
Yes—fractions or decimals both work. For exact arithmetic, fractions are preferred.
Vertical lines?
A vertical line $x=c$ has undefined slope and cannot be written as $y=mx+b$.
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Convert to y=mx+b
- m = (y2 − y1)/(x2 − x1)
- y − y1 = m(x − x1) → rearrange to y = mx + b
- Move x-terms and constants, then divide to get y = mx + b