Standard ↔ Slope–Intercept
Algebra, Lines
Intro: Pick a direction and we’ll show the algebra line by line.
Worked example
- Convert $2x + 3y = 9$ to $y=mx+b$.
- Start from standard form: $$2x + 3y = 9.$$
- Isolate the $y$-term by subtracting $2x$ from both sides: $$3y = -2x + 9.$$
- Divide both sides by $3$ to solve for $y$: $$y = -\tfrac{2}{3}x + 3.$$
- Read off parameters: $$m=-\tfrac{2}{3},\quad b=3.$$
- Checks (optional):
- • $y$-intercept at $x=0$: $y=3$ → point $(0,3)$ satisfies $2(0)+3(3)=9$ ✔️.
- • Slope as $-A/B$: $-\frac{2}{3}$ matches $m=-\tfrac{2}{3}$ ✔️.
- Conclusion: $$\boxed{y = -\tfrac{2}{3}x + 3}.$$
- Convert $y = \tfrac{5}{2}x - 4$ to $Ax+By=C$.
- Start from slope–intercept: $$y = \tfrac{5}{2}x - 4.$$
- Clear the denominator by multiplying both sides by $2$: $$2y = 5x - 8.$$
- Move all terms to the left to get $Ax+By=C$: $$5x - 2y = 8.$$
- Optional normalization (make $A>0$ and integer coefficients—already satisfied).
- Quick check: set $x=0$ → $-2y=8$ → $y=-4$, which matches the intercept in the original form ✔️.
- Conclusion: $$\boxed{5x - 2y = 8}.$$
FAQs
Do you simplify fractions?
Yes, we present simplified rational coefficients and prefer $A>0$ in standard form.
Vertical lines?
A vertical line $x=c$ cannot be written as $y=mx+b$ because the slope is undefined; its standard form is $x=c$.
Why choose Mathix?
- Get clear, step-by-step solutions that explain the “why,” not just the answer.
- See the rules used at each step (power, product, quotient, chain, and more).
- Optional animated walk-throughs to make tricky ideas click faster.
- Clean LaTeX rendering for notes, homework, and study guides.
Standard ↔ Slope–Intercept
- From Ax+By=C, subtract Ax and divide by B to get y = (−A/B)x + C/B
- From y=mx+b with fractional m,b, multiply both sides to eliminate denominators
- Move terms to one side, prefer integer coefficients and A>0