Determine whether each pair of lines is parallel,...

Determine whether each pair of lines is parallel, perpendicular, or neither. Verify by graphing on your calculator using a square window. [ See Calculator Note 11A. ( $\hbar$ ) a. $y=0.8(x-4)+7$ b. $y=5-2 x$ $y=-1.25(x-3)+1$ $y=-4+2 x$ c. $6 x-4 y=11$ d. $3 x+2 y=12$ $-9 x+6 y=-4$ $10 x-15 y=7$

Determine whether each pair of lines is parallel, perpendicular, or neither. Verify by graphing on your calculator using a square window. [ See Calculator Note 11A. ( $\hbar$ )
a. $y=0.8(x-4)+7$
b. $y=5-2 x$
$y=-1.25(x-3)+1$
$y=-4+2 x$
c. $6 x-4 y=11$
d. $3 x+2 y=12$
$-9 x+6 y=-4$
$10 x-15 y=7$

Key Concepts

Slope-Intercept Form

This is a way of writing the equation of a line as y = mx + b, where m represents the slope and b represents the y-intercept. Converting a line's equation to this form makes it easier to identify the line’s slope, which is crucial for determining the relationship between lines, such as whether they are parallel or perpendicular.

Slope

The slope of a line is a measure of its steepness, calculated as the ratio of the vertical change to the horizontal change between two points on the line. In the context of comparing lines, slopes provide a direct basis for determining whether lines are parallel (equal slopes) or perpendicular (slopes are negative reciprocals).

Parallel Lines

Parallel lines are defined as lines in the same plane that never intersect. In algebraic terms, this is achieved when two lines have the same slope but different y-intercepts. Recognizing parallelism is important when analyzing the geometric relationships between lines.

Perpendicular Lines

Perpendicular lines intersect at a right angle (90 degrees). Algebraically, two lines are perpendicular if the product of their slopes is -1, meaning each slope is the negative reciprocal of the other. This relationship is a key concept when determining the orientation of intersecting lines.

Graphical Verification

Graphing functions is a practical method to visually confirm relationships between lines such as parallelism or perpendicularity. By plotting the lines on a coordinate grid—often using a calculator with a square window to ensure accurate proportions—you can observe their intersection points (or lack thereof) and verify their slopes visually.

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