The equation of a line is given below. Find the slope of a line that is a. parallel to the line with the given equation, and b. perpendicular to the line with the given equation. $$ y=4 x $$ a. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. A line parallel to the given line has slope $$ \square $$.(Simplify your answer.) 14 of 20 , This question: 1 point(s) possible B. The slope is undefined. b. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. A line perpendicular to the given line has slope $$ \square $$. (Simplify your answer.) B. The slope is undefined.

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UpStudy AI · Accepted Answer

Let's analyze the given equation and find the required slopes.

**Given Equation:**
$$ y = 4x $$

This equation is in the slope-intercept form $$ y = mx + b $$, where $$ m $$ is the slope and $$ b $$ is the y-intercept.

Here, the **slope ($$ m $$)** of the given line is **4**.

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### a. Slope of a Line Parallel to the Given Line

**Parallel Lines:** Lines that are parallel to each other have **identical slopes**.

**Choice A:**  
A line parallel to the given line has slope $$ \boxed{4} $$.

**Choice B:**  
A slope being undefined implies a vertical line, which is not parallel to the given line.

**Correct Choice:** **A.** A line parallel to the given line has slope $$ \boxed{4} $$.

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### b. Slope of a Line Perpendicular to the Given Line

**Perpendicular Lines:** The slopes of two perpendicular lines are **negative reciprocals** of each other. If the slope of one line is $$ m $$, the slope of a perpendicular line is $$ -\frac{1}{m} $$.

Given the slope $$ m = 4 $$, the slope of a perpendicular line is:
$$ -\frac{1}{4} $$

**Choice A:**  
A line perpendicular to the given line has slope $$ \boxed{-\frac{1}{4}} $$.

**Choice B:**  
A slope being undefined implies a vertical line, which is not necessarily perpendicular to the given line unless the given line is horizontal (which it isn't in this case).

**Correct Choice:** **A.** A line perpendicular to the given line has slope $$ \boxed{-\frac{1}{4}} $$.

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**Summary:**
- **a.** Parallel slope: **4**
- **b.** Perpendicular slope: **-$$\frac{1}{4}$$**
a. A line parallel to the given line has slope **4**. b. A line perpendicular to the given line has slope **-$$\frac{1}{4}$$**.

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