Understanding Linear Functions: Graphing and Solving Equations in Two Variables

Linear Functions Basics

Linear equations in two variables always follow the form Ax + By = C, where A, B, and C are real numbers (and A and B can't both be zero). When you solve these equations, you're looking for ordered pairs (x,y) that make the equation true.

To graph linear equations, you can find key points like the x-intercept (where the line crosses the x-axis) and y-intercept (where it crosses the y-axis). For example, in the equation 2x+8y=12, the y-intercept is (0,1.5) and the x-intercept is (6,0).

Slope is a critical concept that tells you how steep a line is. The formula is m = $y₂-y₁$/$x₂-x₁$. You'll use different forms of linear equations depending on what information you have:

• Slope-intercept form: y = mx + b
• Point-slope form: y - y₁ = m$x - x₁$

Pro Tip: When solving linear inequalities $like y > 3x-2$, remember that the line is solid for ≤ or ≥, and dashed for < or >. The shading tells you which points satisfy the inequality - shade upward or right for "greater than" relationships!