Question
Put the equation into slope-intercept form:
2y + 4x = 12
5x - y = 9
What form is this equation in?
What do you know in this form?
Can you graph it easily?
Find the slope of a line perpendicular to each given line.
What does a perpendicular slope look like?
Y = -1/2x - 2
y + 7 = 5/2 (x + 6)
What if we wanted a parallel line? What does the slope of a parallel line look like?
Find the slope, y-intercept, and x-intercept for each equation.
y - 4 = 5(x + 2)
8x - 7y = 17
2y + 4x = 12
5x - y = 9
What form is this equation in?
What do you know in this form?
Can you graph it easily?
Find the slope of a line perpendicular to each given line.
What does a perpendicular slope look like?
Y = -1/2x - 2
y + 7 = 5/2 (x + 6)
What if we wanted a parallel line? What does the slope of a parallel line look like?
Find the slope, y-intercept, and x-intercept for each equation.
y - 4 = 5(x + 2)
8x - 7y = 17
Answers
GPT-4o mini
### 1. Convert the equation into slope-intercept form
**Equation:**
\[ 2y + 4x = 12 \]
**Steps to convert to slope-intercept form (y = mx + b):**
1. Subtract \(4x\) from both sides:
\[ 2y = -4x + 12 \]
2. Divide each term by 2:
\[ y = -2x + 6 \]
**Slope-intercept form:**
\[ y = -2x + 6 \]
---
### 2. Equation: \( 5x - y = 9 \)
**What form is this equation in?**
This equation is in standard form (Ax + By = C).
**What do you know in this form?**
In the standard form, you can easily find intercepts, and it clearly shows the relationship between the coefficients (A, B, C).
**Can you graph it easily?**
It’s not as straightforward as slope-intercept form, as you first need to rearrange it to find the slope and y-intercept.
**Convert the equation to slope-intercept form:**
1. Add \(y\) to both sides:
\[ 5x - 9 = y \]
2. Rewrite:
\[ y = 5x - 9 \]
---
### 3. Find the slope of a line perpendicular to each given line
To find the perpendicular slope, take the negative reciprocal of the original slope.
**For \(y = -2x + 6\):**
- The slope \(m = -2\).
- The perpendicular slope \(m_{\perpendicular} = \frac{1}{2}\).
**For \(y = 5x - 9\):**
- The slope \(m = 5\).
- The perpendicular slope \(m_{\perpendicular} = -\frac{1}{5}\).
---
### 4. What does a perpendicular slope look like?
A perpendicular slope is obtained by flipping the fraction of the slope and changing its sign.
---
### 5. Given two equations:
- **Equation 1:**
\[ y = -\frac{1}{2}x - 2 \]
- Slope \(m = -\frac{1}{2}\)
- Perpendicular slope \(m_{\perpendicular} = 2\)
- **Equation 2:**
\[ y + 7 = \frac{5}{2}(x + 6) \]
- Rewrite to slope-intercept form:
\[ y + 7 = \frac{5}{2}x + 15 \]
\[ y = \frac{5}{2}x + 8 \]
- Slope \(m = \frac{5}{2}\)
- Perpendicular slope \(m_{\perpendicular} = -\frac{2}{5}\)
---
### 6. What if we wanted a parallel line?
The slope of a parallel line is the same as the original line.
**Examples of parallel slopes:**
- For the line \(y = -2x + 6\), a parallel line will also have a slope of \(-2\).
- For the line \(y = 5x - 9\), a parallel line will also have a slope of \(5\).
- For the line \(y = -\frac{1}{2}x - 2\), a parallel line will also have a slope of \(-\frac{1}{2}\).
- For the line \(y = \frac{5}{2}x + 8\), a parallel line will also have a slope of \(\frac{5}{2}\).
---
### 7. Find the slope, y-intercept, and x-intercept for each equation
1. **Equation:**
\[ y - 4 = 5(x + 2) \]
**Convert to slope-intercept form:**
\[ y - 4 = 5x + 10 \]
\[ y = 5x + 14 \]
- **Slope (m):** 5
- **y-intercept (b):** 14 (at (0, 14))
- **x-intercept:** Set \(y = 0\):
\[ 0 = 5x + 14 \]
\[ 5x = -14 \]
\[ x = -\frac{14}{5} \] (at \((-2.8, 0)\))
---
2. **Equation:**
\[ 8x - 7y = 17 \]
**Convert to slope-intercept form:**
\[ -7y = -8x + 17 \]
\[ y = \frac{8}{7}x - \frac{17}{7} \]
- **Slope (m):** \(\frac{8}{7}\)
- **y-intercept (b):** \(-\frac{17}{7}\) (at \((0, -\frac{17}{7})\))
- **x-intercept:** Set \(y = 0\):
\[ 0 = 8x - 17 \]
\[ 8x = 17 \]
\[ x = \frac{17}{8} \] (at \((\frac{17}{8}, 0)\))
---
These steps provide you with all the information regarding the slopes, intercepts, and how to determine parallel and perpendicular lines.
**Equation:**
\[ 2y + 4x = 12 \]
**Steps to convert to slope-intercept form (y = mx + b):**
1. Subtract \(4x\) from both sides:
\[ 2y = -4x + 12 \]
2. Divide each term by 2:
\[ y = -2x + 6 \]
**Slope-intercept form:**
\[ y = -2x + 6 \]
---
### 2. Equation: \( 5x - y = 9 \)
**What form is this equation in?**
This equation is in standard form (Ax + By = C).
**What do you know in this form?**
In the standard form, you can easily find intercepts, and it clearly shows the relationship between the coefficients (A, B, C).
**Can you graph it easily?**
It’s not as straightforward as slope-intercept form, as you first need to rearrange it to find the slope and y-intercept.
**Convert the equation to slope-intercept form:**
1. Add \(y\) to both sides:
\[ 5x - 9 = y \]
2. Rewrite:
\[ y = 5x - 9 \]
---
### 3. Find the slope of a line perpendicular to each given line
To find the perpendicular slope, take the negative reciprocal of the original slope.
**For \(y = -2x + 6\):**
- The slope \(m = -2\).
- The perpendicular slope \(m_{\perpendicular} = \frac{1}{2}\).
**For \(y = 5x - 9\):**
- The slope \(m = 5\).
- The perpendicular slope \(m_{\perpendicular} = -\frac{1}{5}\).
---
### 4. What does a perpendicular slope look like?
A perpendicular slope is obtained by flipping the fraction of the slope and changing its sign.
---
### 5. Given two equations:
- **Equation 1:**
\[ y = -\frac{1}{2}x - 2 \]
- Slope \(m = -\frac{1}{2}\)
- Perpendicular slope \(m_{\perpendicular} = 2\)
- **Equation 2:**
\[ y + 7 = \frac{5}{2}(x + 6) \]
- Rewrite to slope-intercept form:
\[ y + 7 = \frac{5}{2}x + 15 \]
\[ y = \frac{5}{2}x + 8 \]
- Slope \(m = \frac{5}{2}\)
- Perpendicular slope \(m_{\perpendicular} = -\frac{2}{5}\)
---
### 6. What if we wanted a parallel line?
The slope of a parallel line is the same as the original line.
**Examples of parallel slopes:**
- For the line \(y = -2x + 6\), a parallel line will also have a slope of \(-2\).
- For the line \(y = 5x - 9\), a parallel line will also have a slope of \(5\).
- For the line \(y = -\frac{1}{2}x - 2\), a parallel line will also have a slope of \(-\frac{1}{2}\).
- For the line \(y = \frac{5}{2}x + 8\), a parallel line will also have a slope of \(\frac{5}{2}\).
---
### 7. Find the slope, y-intercept, and x-intercept for each equation
1. **Equation:**
\[ y - 4 = 5(x + 2) \]
**Convert to slope-intercept form:**
\[ y - 4 = 5x + 10 \]
\[ y = 5x + 14 \]
- **Slope (m):** 5
- **y-intercept (b):** 14 (at (0, 14))
- **x-intercept:** Set \(y = 0\):
\[ 0 = 5x + 14 \]
\[ 5x = -14 \]
\[ x = -\frac{14}{5} \] (at \((-2.8, 0)\))
---
2. **Equation:**
\[ 8x - 7y = 17 \]
**Convert to slope-intercept form:**
\[ -7y = -8x + 17 \]
\[ y = \frac{8}{7}x - \frac{17}{7} \]
- **Slope (m):** \(\frac{8}{7}\)
- **y-intercept (b):** \(-\frac{17}{7}\) (at \((0, -\frac{17}{7})\))
- **x-intercept:** Set \(y = 0\):
\[ 0 = 8x - 17 \]
\[ 8x = 17 \]
\[ x = \frac{17}{8} \] (at \((\frac{17}{8}, 0)\))
---
These steps provide you with all the information regarding the slopes, intercepts, and how to determine parallel and perpendicular lines.