Which of these equations represents a line parallel to y = 4x − 5? y = 4x + 7, y = −(1/4)x + 6, y = (3/4)x + 1, or y = (3/2)x − 5.

Step-by-step solution

1. 1

   State the criterion. Two distinct lines are parallel exactly when they have the same slope and different intercepts. The $y$-intercept plays no role in deciding parallelism, so the entire question reduces to comparing slopes.

2. 2

   Read the slope of the given line. In slope-intercept form $y=mx+b$, the slope is the coefficient of $x$:

   $y=4x-5\quad\Longrightarrow\quad m=4$

3. 3

   Read the slope of each option. All four are already in slope-intercept form:

   $y=4x+7\Rightarrow m=4;\quad y=-\tfrac14x+6\Rightarrow m=-\tfrac14;\quad y=\tfrac34x+1\Rightarrow m=\tfrac34;\quad y=\tfrac32x-5\Rightarrow m=\tfrac32$

4. 4

   Select the match. Only the first has slope $4$:

   $\boxed{y=4x+7}$

5. 5

   Confirm it is parallel and not the same line. Its intercept $7$ differs from $-5$, so the two lines are distinct and never meet — setting $4x-5=4x+7$ gives $-5=7$, a contradiction, confirming no intersection.

6. 6

   Note the deliberate distractor. The option $y=-\tfrac14x+6$ has slope $-\tfrac14$, the negative reciprocal of $4$, so that line is perpendicular to the given line, not parallel. Mixing up $m$ and $-1/m$ is the intended trap.