Algebra and Linear Relationships
Core concepts
A linear relationship has a constant rate of change. The slope-intercept form y = mx + b tells you the slope m (change in y per unit change in x) and the y-intercept b. The point-slope form y − y₁ = m(x − x₁) is useful when you know a point and the slope. To find a line through two points, compute m = (y₂ − y₁)/(x₂ − x₁), then substitute one point to solve for b. Solving equations means isolating the variable by inverse operations, always applying the same operation to both sides. For inequalities, the key rule is that multiplying or dividing by a negative number reverses the inequality sign. Systems of two linear equations can be solved by substitution or elimination; the solution is the intersection point.
Traps to expect
- Slope sign errors when subtracting coordinates; keep the order consistent (y₂ − y₁ over x₂ − x₁).
- Forgetting to flip the inequality sign after dividing by a negative.
- Confusing horizontal (slope 0) with vertical (undefined slope) lines.
- Distributing a negative sign incorrectly, e.g. −(x − 3) = −x + 3, not −x − 3.
- Reporting a system's solution as one number when it needs both x and y.
Worked example
Find the line through (2, 5) and (6, 13). Slope m = (13 − 5)/(6 − 2) = 8/4 = 2. Using y = mx + b with (2, 5): 5 = 2(2) + b, so b = 1. The equation is y = 2x + 1. Check with the other point: 2(6) + 1 = 13. ✓
Rule of thumb
Always substitute your answer back into the original equation, and check that a slope makes sense: positive slope rises left to right, negative slope falls.