ACT Math: Find Slope and Write Linear Equations from Two Points

Slope is the rate of change: how much y changes for each unit x changes. Formula: slope=m=(y2-y1)/(x2-x1). Example: Points (2,3) and (5,9). Slope=m=(9-3)/(5-2)=6/3=2. This means for every 1 unit increase in x, y increases by 2 units. The slope formula is mechanical; plug in coordinates and simplify. Most errors come from mixing up the order (y on top, x on bottom) or arithmetic mistakes.

Negative slope example: Points (1,8) and (4,2). Slope=m=(2-8)/(4-1)=-6/3=-2. Negative slope means as x increases, y decreases. Zero slope means horizontal line (y doesn't change). Undefined slope means vertical line (x doesn't change, so we divide by zero).

Step 1: Calculate slope using m=(y2-y1)/(x2-x1). Step 2: Use point-slope form: y-y1=m(x-x1). Plug in slope and one point. Step 3: Simplify to slope-intercept form: y=mx+b. Example: Find the line through (1,2) and (3,6). Step 1: m=(6-2)/(3-1)=4/2=2. Step 2: y-2=2(x-1). Step 3: y-2=2x-2, so y=2x. The equation is y=2x. This three-step process works for every linear equation question on ACT Math.

Practice: Find the line through (2,5) and (4,9). Step 1: m=(9-5)/(4-2)=4/2=2. Step 2: y-5=2(x-2). Step 3: y-5=2x-4, so y=2x+1. Verify: At x=2, y=2(2)+1=5. ✓ At x=4, y=2(4)+1=9. ✓

Mistake 1: Flipping numerator and denominator (putting x on top). Fix: Always y on top, x on bottom. Mistake 2: Subtracting in the wrong order (using y1-y2 instead of y2-y1). Fix: It doesn't matter as long as you're consistent, but choose one point and stick with it. Mistake 3: Arithmetic errors when simplifying the fraction. Fix: Simplify slowly; don't rush. Mistake 4: Forgetting to simplify y=mx+b to standard form if required. Fix: Check what form the question asks for. These four mistakes cause 90% of slope and linear equation errors.

Drill: Find the slope for each pair. (1) (0,0) and (2,4). Answer: m=2. (2) (1,5) and (3,1). Answer: m=-2. (3) (2,3) and (2,7). Answer: undefined (vertical line). (4) (1,4) and (5,4). Answer: m=0 (horizontal line). Work through these carefully.

Slope appears in its own questions, in line-writing questions, in parallel/perpendicular line questions, and in distance/rate problems. Mastering the basic formula and three-step method gives you a foundation for all these. Students who solve slope questions mechanically never stumble on more complex topics that build on slope.

Spend one week solving ten slope problems a day using the three-step method. By test day, finding slope and writing equations will be reflex-level automatic. That speed and accuracy will compound across related question types.