Question
Graph the following system of linear equations, then answer the questions below the graph.
y=3/4x+2
4x+3y=0
Part A: Graph the following system of linear equations.
Use the drop-down menus to complete the statements.
The two lines are [ parallel, perpendicular, intersecting but not perpendicular] The system of linear equations is [independent, consistent, inconsistent] which means it [does, does not] have a solution.
also, please help me plot the lines and tell me where.
y=3/4x+2
4x+3y=0
Part A: Graph the following system of linear equations.
Use the drop-down menus to complete the statements.
The two lines are [ parallel, perpendicular, intersecting but not perpendicular] The system of linear equations is [independent, consistent, inconsistent] which means it [does, does not] have a solution.
also, please help me plot the lines and tell me where.
Answers
GPT-4o mini
To graph the system of equations and analyze their relationships, let’s start by rewriting them in a usable form.
1. **First equation:**
\( y = \frac{3}{4}x + 2 \)
This is already in slope-intercept form \( y = mx + b \), where the slope \( m = \frac{3}{4} \) and the y-intercept \( b = 2 \).
2. **Second equation:**
\( 4x + 3y = 0 \)
To convert this to slope-intercept form, solve for \( y \):
\[
3y = -4x \\
y = -\frac{4}{3}x
\]
Now, we can see that the slope \( m = -\frac{4}{3} \) and the y-intercept \( b = 0 \).
### Plotting the Lines:
1. **Graphing \( y = \frac{3}{4}x + 2 \)**:
- The y-intercept is \( 2 \) (point \( (0, 2) \)).
- The slope \( \frac{3}{4} \) means that for every 4 units you move to the right (positive x-direction), you move up 3 units (positive y-direction).
- Another point can be found using \( x = 4 \):
\[
y = \frac{3}{4}(4) + 2 = 3 + 2 = 5 \quad \text{(point (4, 5))}
\]
2. **Graphing \( y = -\frac{4}{3}x \)**:
- The y-intercept is \( 0 \) (point \( (0, 0) \)).
- The slope \( -\frac{4}{3} \) means that for every 3 units you move to the right, you move down 4 units (negative y-direction).
- Another point can be found using \( x = 3 \):
\[
y = -\frac{4}{3}(3) = -4 \quad \text{(point (3, -4))}
\]
### Analysis of Lines:
- The first line has a slope of \( \frac{3}{4} \).
- The second line has a slope of \( -\frac{4}{3} \).
Since the slopes are negative reciprocals of each other, the lines are **perpendicular**.
### Conclusion:
- The two lines are **perpendicular**.
- The system of linear equations is **independent** (since the lines intersect at a single point).
- This means it **does** have a solution.
### Summary:
- **Complete statements:**
- The two lines are **perpendicular**.
- The system of linear equations is **independent**.
- This means it **does** have a solution.
For plotting:
- Line 1: Points at (0, 2) and (4, 5)
- Line 2: Points at (0, 0) and (3, -4)
Use these coordinates to graph the lines accordingly on your graphing tool or paper. The intersection of these two lines will give you the solution to the system, which can be found by solving the equations simultaneously.
1. **First equation:**
\( y = \frac{3}{4}x + 2 \)
This is already in slope-intercept form \( y = mx + b \), where the slope \( m = \frac{3}{4} \) and the y-intercept \( b = 2 \).
2. **Second equation:**
\( 4x + 3y = 0 \)
To convert this to slope-intercept form, solve for \( y \):
\[
3y = -4x \\
y = -\frac{4}{3}x
\]
Now, we can see that the slope \( m = -\frac{4}{3} \) and the y-intercept \( b = 0 \).
### Plotting the Lines:
1. **Graphing \( y = \frac{3}{4}x + 2 \)**:
- The y-intercept is \( 2 \) (point \( (0, 2) \)).
- The slope \( \frac{3}{4} \) means that for every 4 units you move to the right (positive x-direction), you move up 3 units (positive y-direction).
- Another point can be found using \( x = 4 \):
\[
y = \frac{3}{4}(4) + 2 = 3 + 2 = 5 \quad \text{(point (4, 5))}
\]
2. **Graphing \( y = -\frac{4}{3}x \)**:
- The y-intercept is \( 0 \) (point \( (0, 0) \)).
- The slope \( -\frac{4}{3} \) means that for every 3 units you move to the right, you move down 4 units (negative y-direction).
- Another point can be found using \( x = 3 \):
\[
y = -\frac{4}{3}(3) = -4 \quad \text{(point (3, -4))}
\]
### Analysis of Lines:
- The first line has a slope of \( \frac{3}{4} \).
- The second line has a slope of \( -\frac{4}{3} \).
Since the slopes are negative reciprocals of each other, the lines are **perpendicular**.
### Conclusion:
- The two lines are **perpendicular**.
- The system of linear equations is **independent** (since the lines intersect at a single point).
- This means it **does** have a solution.
### Summary:
- **Complete statements:**
- The two lines are **perpendicular**.
- The system of linear equations is **independent**.
- This means it **does** have a solution.
For plotting:
- Line 1: Points at (0, 2) and (4, 5)
- Line 2: Points at (0, 0) and (3, -4)
Use these coordinates to graph the lines accordingly on your graphing tool or paper. The intersection of these two lines will give you the solution to the system, which can be found by solving the equations simultaneously.