This section provides step-by-step examples demonstrating how to apply linear equation concepts. You will learn how to find slopes, write equations in different forms, determine geometric relationships between lines, and calculate distances and midpoints.

Example 1: Slope-Intercept Form to Standard Form (Basic)

Problem: Convert the equation y=−23x+4y = -\frac{2}{3}x + 4 into Standard Form (Ax+By=CAx + By = C, where A,BA, B, and CC are integers).

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Example 2: Point-Slope to Slope-Intercept Form (Intermediate)

Problem: Find the equation of the line that passes through the point (−4,5)(-4, 5) and has a slope of 12\frac{1}{2}. Write the final equation in Slope-Intercept form.

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Example 3: The Distance Formula (Basic)

Problem: Find the distance between the points (2,3)(2, 3) and (5,7)(5, 7).

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Example 4: The Midpoint Formula (Intermediate)

Problem: Find the midpoint of the line segment connecting (−4,2)(-4, 2) and (6,−8)(6, -8).

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Caution

Always be careful with the signs of your coordinates when applying the midpoint or distance formulas. Adding a negative number is equivalent to subtraction. Forgetting parentheses around negative coordinates is a very common source of errors.

Example 5: Equation of a Perpendicular Line (Advanced)

Problem: Find the equation of the line that passes through (6,−1)(6, -1) and is perpendicular to the line 2x+3y=122x + 3y = 12. Write the answer in standard form.

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