Question

Are the lines parallel? $$ y=5+6(x+2) y=5+6(3 x-1) $$

   Are the lines parallel?
$$
y=5+6(x+2) y=5+6(3 x-1)
$$

Algebra Form and Function

Algebra Form and Function

William G. McCallum,… 1st Edition

Chapter 5, Problem 41

Instant Answer

Solved by Expert Nick Johnson

04/05/2022

Step 1

The first equation is $y = 5 + 6(x + 2)$. Distributing the 6 through the parentheses, we get $y = 5 + 6x + 12$. Simplifying, we get $y = 6x + 17$. The second equation is $y = 5 + 6(3x - 1)$. Distributing the 6 through the parentheses, we get $y = 5 + 18x - 6$. Show more…

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Are the lines parallel? $$ y=5+6(x+2) y=5+6(3 x-1) $$

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Key Concepts

Parallel Lines

Parallel lines are lines in a plane that never intersect. This characteristic is guaranteed when the lines have identical slopes. Therefore, comparing the slopes derived from the slope-intercept form of their equations is the fundamental method for establishing if two lines are parallel.

Algebraic Manipulation

Algebraic manipulation involves simplifying expressions by distributing factors, combining like terms, and rewriting equations in standard forms. This process is crucial for converting given equations into the slope-intercept form, which facilitates the extraction of slope values for further analysis.

Slope-Intercept Form

The slope-intercept form is a way of writing a linear equation as y = mx + b, where m represents the slope and b represents the y-intercept. It provides a clear picture of the line's steepness and where it crosses the y-axis, which is essential for analyzing and comparing different lines.

Slope

The slope of a line measures its steepness and direction, defined as the ratio of the change in y to the change in x between two points. Knowing the slope allows you to understand the behavior of the line and is a key factor when determining relationships between lines, such as parallelism.

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