Slope-Intercept Form
y=mx+b. m=slope, b=y-intercept. From two points: m=(yโโyโ)/(xโโxโ), b=yโโmxโ. From Ax+By=C: y=โ(A/B)x+C/B. x-intercept: set y=0, x=โb/m.
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b is the y-intercept: point (0,b) where line crosses y-axis. Parallel lines: same m. Perpendicular: mโ=โ1/mโ. Vertical line x=c has no slope-intercept form (infinite slope).
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Why: Slope-intercept form y=mx+b is ideal for graphing โ slope m and y-intercept b are explicit. Used in linear regression, economics, and physics. Easy to convert from other forms.
How: Given m and b: y=mx+b. Two points: m=(yโโyโ)/(xโโxโ), then b=yโโmxโ. Standard Ax+By=C: solve for y, y=โ(A/B)x+C/B (Bโ 0). x-intercept: 0=mx+b โ x=โb/m.
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Slope & Intercepts
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๐งฎ Fascinating Math Facts
y=2x+3: slope 2, y-intercept (0,3). x-intercept (โ3/2,0).
โ Example
Standard 2x+3y=6 โ y=โ(2/3)x+2. Slope โ2/3, intercept 2.
โ Conversion
๐ Slope-Intercept Form Advantages
The form y = mx + b is ideal for graphing: m gives the slope (rise/run), b gives the y-intercept (0, b). Start at (0, b), then move right 1 and up m (or down if m is negative) to plot another point.
๐ Graphing Linear Equations
To graph y = mx + b: (1) Plot (0, b). (2) Use slope m = rise/run: from (0,b), move right 1, up m. (3) Draw the line through the points.
โ๏ธ Parallel and Perpendicular Lines
Parallel lines have the same slope (mโ = mโ). Perpendicular lines have slopes that are negative reciprocals: mโ = -1/mโ.
๐ Real-World Linear Models
Cost = fixed + variableยทquantity (C = b + mx). Revenue, profit, distance = rateรtime, and many physics/economic relationships are linear.
โ ๏ธ Common Mistakes
- Vertical lines (x = k) have undefined slope โ cannot write as y = mx + b.
- When converting Ax+By=C to y=mx+b, remember m = -A/B and b = C/B.
- Perpendicular slope is -1/m, not 1/m.
๐ Worked Examples
๐ Summary
Slope-intercept form y = mx + b is ideal for graphing and modeling. Convert from two points using m = (yโ-yโ)/(xโ-xโ) and b = yโ - mยทxโ. From Ax+By=C, use m = -A/B and b = C/B. Parallel lines share slope; perpendicular slopes multiply to -1.
โ FAQ
What is slope-intercept form?
y = mx + b where m is slope and b is y-intercept. The most convenient form for graphing.
How do I find slope from two points?
m = (yโ - yโ)/(xโ - xโ). Then b = yโ - mยทxโ.
How do I convert Ax+By=C to y=mx+b?
Solve for y: y = (-A/B)x + (C/B), so m = -A/B and b = C/B.
What is the perpendicular slope?
mโฅ = -1/m. If m = 2, perpendicular slope is -1/2.
When is slope undefined?
Vertical lines (x = k) have undefined slope. Cannot write as y = mx + b.
What does horizontal line look like?
y = b (m = 0). Flat line, no x-intercept unless b = 0.
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