The equations of three lines are giwen belous.

Line
Line
Line 3:4x+16y=-16
For each pair of lines, determine whether they are paraliel, perpendicular, or meithet

Line 1 and Line 2 - Parallel Perpendicular Neither

Line 1 and Line 3: Parallel Perpendicular Nether

Line 2 and Line 3 : Paraliel Perpendicular Neither

Ask by Gordon Luna. in the United States

Mar 31,2025

Question

The equations of three lines are giwen belous.

Line
Line
Line 3:4x+16y=-16
For each pair of lines, determine whether they are paraliel, perpendicular, or meithet

Line 1 and Line 2 - Parallel Perpendicular Neither

Line 1 and Line 3: Parallel Perpendicular Nether

Line 2 and Line 3 : Paraliel Perpendicular Neither

Ask by Gordon Luna. in the United States

Mar 31,2025

UpStudy AI · Accepted Answer

To determine the relationship between the lines, we need to find the slopes of each line.

1. **Line 1:** The equation is given as $$ y = -4x - 5 $$. The slope $$ m_1 $$ is -4.
2. **Line 2:** The equation is given as $$ zy = \frac{1}{4}x + 2 $$. We can rewrite it in slope-intercept form $$ y = mx + b $$ by isolating $$ y $$:
   $$
   y = \frac{1}{4z}x + \frac{2}{z}
   $$
   The slope $$ m_2 $$ is $$ \frac{1}{4z} $$.
3. **Line 3:** The equation is $$ 4x + 16y = -16 $$. We can rewrite it in slope-intercept form:
   $$
   16y = -4x - 16 \implies y = -\frac{1}{4}x - 1
   $$
   The slope $$ m_3 $$ is $$ -\frac{1}{4} $$.

Now we will compare the slopes to determine if the lines are parallel, perpendicular, or neither.

1. **Line 1 and Line 2:**
   - $$ m_1 = -4 $$
   - $$ m_2 = \frac{1}{4z} $$
   - Lines are parallel if $$ m_1 = m_2 $$, perpendicular if $$ m_1 \cdot m_2 = -1 $$, and neither otherwise.

2. **Line 1 and Line 3:**
   - $$ m_1 = -4 $$
   - $$ m_3 = -\frac{1}{4} $$
   - Check for parallelism and perpendicularity.

3. **Line 2 and Line 3:**
   - $$ m_2 = \frac{1}{4z} $$
   - $$ m_3 = -\frac{1}{4} $$
   - Check for parallelism and perpendicularity.

Now we will analyze each pair:

1. **Line 1 and Line 2:**
   - For parallel: $$ -4 = \frac{1}{4z} $$ implies $$ z = -\frac{1}{16} $$.
   - For perpendicular: $$ -4 \cdot \frac{1}{4z} = -1 $$ implies $$ z = -\frac{1}{16} $$.
   - Conclusion: **Neither** (unless $$ z = -\frac{1}{16} $$).

2. **Line 1 and Line 3:**
   - Check for perpendicular: $$ -4 \cdot -\frac{1}{4} = 1 $$.
   - Conclusion: **Perpendicular**.

3. **Line 2 and Line 3:**
   - For parallel: $$ \frac{1}{4z} = -\frac{1}{4} $$ implies $$ z = -1 $$.
   - For perpendicular: $$ \frac{1}{4z} \cdot -\frac{1}{4} = -1 $$ implies $$ z = -\frac{1}{4} $$.
   - Conclusion: **Neither** (unless specific values of $$ z $$).

- Line 1 and Line 2: Neither - Line 1 and Line 3: Perpendicular - Line 2 and Line 3: Neither

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