Topic A: Linear Equations & Graphing
Lessons 1–4 · Slope-intercept, standard, and point-slope forms.
01–02 — Lessons 1–2
Forms of Linear Equations
- Slope-intercept: y = mx + b. Slope m, y-intercept b.
- Point-slope: y − y₁ = m(x − x₁). Use when given slope and a point.
- Standard form: Ax + By = C. Useful for x and y intercepts.
- Slope: m = rise/run = (y₂ − y₁)/(x₂ − x₁).
- Parallel lines: same slope. Perpendicular: slopes are negative reciprocals.
Through (1, 3) with slope 2: y − 3 = 2(x − 1) → y = 2x + 1
Practice — Lessons 1–2
Q1 · Short Answer
What is the slope of the line through (2, 5) and (6, 13)?
Q2 · Multiple Choice
Two lines are perpendicular. One has slope 3. What is the other's slope?
03–04 — Lessons 3–4
Graphing Linear Equations
- Plot the y-intercept, then use slope (rise over run) to find a second point.
- For standard form Ax + By = C: find x-intercept (set y=0) and y-intercept (set x=0).
- Horizontal lines: y = k (slope 0). Vertical lines: x = h (undefined slope).
- A line with positive slope goes up left to right. Negative slope goes down.
y = −(2/3)x + 4 → start at (0,4), go right 3 and down 2 to next point (3,2).
Practice — Lessons 3–4
Q3 · Multiple Choice
The x-intercept of 3x + 2y = 12 is:
Q4 · Multiple Choice
A horizontal line has slope:
Topic B: Systems of Linear Equations
Lessons 5–9 · Solving systems by graphing, substitution, and elimination.
05–07 — Lessons 5–7
Solving Systems
- Graphing: intersection point is the solution.
- Substitution: isolate one variable, substitute into the other equation.
- Elimination: add/subtract equations to cancel one variable.
- One solution: lines intersect. No solution: parallel (same slope, different intercepts). Infinite: same line.
y = 3x − 1 and y = x + 5 → 3x−1 = x+5 → 2x = 6 → x=3, y=8. Solution: (3,8)
Practice — Lessons 5–7
Q5 · Short Answer
Solve by substitution: y = 2x and 3x + y = 15. What is x?
Q6 · Multiple Choice
A system has no solution. This means the lines are:
08–09 — Lessons 8–9
Linear Inequalities & Systems of Inequalities
- Linear inequality: like an equation but with <, >, ≤, or ≥.
- Graph: shade the half-plane. Solid line for ≤ or ≥, dashed for < or >.
- System of inequalities: shade both — the overlapping region is the solution.
- Test a point (e.g., origin) to check which side to shade.
y > 2x − 1: dashed line, shade above. y ≤ x + 3: solid line, shade below. Overlap is the solution.
Practice — Lessons 8–9
Q7 · Multiple Choice
For y < 3x + 2, which type of boundary line do you use?
Q8 · Multiple Choice
The solution to a system of linear inequalities is: