Question

Are the lines parallel? $$ 2 x+3 y=5 ; 4 x+6 y=7 $$

   Are the lines parallel?
$$
2 x+3 y=5 ; 4 x+6 y=7
$$

Algebra Form and Function

Algebra Form and Function

William G. McCallum,… 1st Edition

Chapter 5, Problem 38

Instant Answer

Solved by Expert Nick Johnson

04/05/2022

Step 1

The first equation is $2x + 3y = 5$. We can rewrite this as $y = -\frac{2}{3}x + \frac{5}{3}$. Show more…

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Are the lines parallel? $$ 2 x+3 y=5 ; 4 x+6 y=7 $$

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Key Concepts

Coefficient Proportionality

When comparing lines in standard form, the proportionality of the coefficients of the x and y terms is indicative of parallelism. If the ratios of the coefficients of x and y in one equation are the same as those in another equation, the lines are candidates for being parallel provided the constant terms are not in the same proportion.

Slope of a Line

The slope of a line describes its steepness and direction. In the context of linear equations, the slope can be calculated by rewriting the equation in slope-intercept form or by comparing the coefficients from the standard form. Recognizing the slope is essential when determining relationships between lines such as whether they are parallel or not.

Criteria for Parallel Lines

Two lines are parallel if and only if they have the same slope and different y-intercepts. This means that after the equations are converted into slope-intercept form, if the slopes match while the constant terms differ, the lines will never intersect, thereby being parallel.

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