Line Through Two Points
Algebra, Lines
Intro: We compute m, write point–slope, then simplify to y=mx+b.
Worked example
- Line through $(1,-3)$ and $(4,6)$
- Identify points: $A(x_1,y_1)=(1,-3)$ and $B(x_2,y_2)=(4,6)$.
- Slope formula: $$m=\frac{y_2-y_1}{x_2-x_1}.$$
- Compute numerator: $y_2-y_1=6-(-3)=9$.
- Compute denominator: $x_2-x_1=4-1=3$.
- Therefore slope: $$m=\frac{9}{3}=3.$$
- Point–slope form with point $A(1,-3)$: $$y-y_1=m(x-x_1)\;\Rightarrow\; y-(-3)=3\,(x-1).$$
- Simplify the left: $y+3=3(x-1)=3x-3$.
- Isolate $y$: $$y=3x-3-3=3x-6.$$
- Slope–intercept form: $$\boxed{y=3x-6}.$$
- Check with point $B(4,6)$: plug $x=4$ into $y=3x-6$ to get $y=12-6=6$ ✔️.
- Intercepts (optional):
- • $y$-intercept: set $x=0$ → $y=-6$ so intercept is $(0,-6)$.
- • $x$-intercept: set $y=0$ → $0=3x-6$ → $x=2$ so intercept is $(2,0)$.
FAQs
Vertical line?
If $x2 = x1$, the slope is undefined. The equation is vertical: $x = constant$ (specifically $x = x1$).
Order of points matters?
No—the same line results. Swapping the points changes numerator and denominator signs but the slope value is unchanged.
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How this calculator works
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\sin,\lnworks too). - Press Generate a practice question button to generate the derivative and the full reasoning.
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