Question
Consider the system of equations.
y = 3x + 2 y = −
2
3
x − 4
Explain why these particular equations can be graphed immediately.
y = 3x + 2 y = −
2
3
x − 4
Explain why these particular equations can be graphed immediately.
Answers
Answered by
GPT-4o mini
The system of equations provided is:
1. \( y = 3x + 2 \)
2. \( y = -\frac{2}{3} x - 4 \)
These equations are in the slope-intercept form of a linear equation, which is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. This form allows for immediate graphing for two main reasons:
1. **Identifying the Slope and Y-intercept**: Each equation provides a clear slope and y-intercept:
- For the first equation \( y = 3x + 2 \), the slope \( m \) is 3, and the y-intercept \( b \) is 2. This means that the line crosses the y-axis at (0, 2), and for every 1 unit increase in \( x \), \( y \) increases by 3 units.
- For the second equation \( y = -\frac{2}{3} x - 4 \), the slope \( m \) is \(-\frac{2}{3}\), and the y-intercept \( b \) is -4. This means that the line crosses the y-axis at (0, -4), and for every 1 unit increase in \( x \), \( y \) decreases by \(\frac{2}{3}\) units.
2. **Quickly Plotting Points**: Since both equations are in slope-intercept form, plotting the lines can be done quickly:
- Start by plotting the y-intercept of each line on the graph.
- Use the slope to find a second point. For the first line, from (0, 2), you can move up 3 units and right 1 unit to find the point (1, 5), for example. For the second line, from (0, -4), move down 2 units and right 3 units to find the point (3, -6).
- Draw the lines through the plotted points.
This explicit structure of the equations enables a straightforward approach to plotting and visualizing the system of equations immediately without additional manipulation.
1. \( y = 3x + 2 \)
2. \( y = -\frac{2}{3} x - 4 \)
These equations are in the slope-intercept form of a linear equation, which is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. This form allows for immediate graphing for two main reasons:
1. **Identifying the Slope and Y-intercept**: Each equation provides a clear slope and y-intercept:
- For the first equation \( y = 3x + 2 \), the slope \( m \) is 3, and the y-intercept \( b \) is 2. This means that the line crosses the y-axis at (0, 2), and for every 1 unit increase in \( x \), \( y \) increases by 3 units.
- For the second equation \( y = -\frac{2}{3} x - 4 \), the slope \( m \) is \(-\frac{2}{3}\), and the y-intercept \( b \) is -4. This means that the line crosses the y-axis at (0, -4), and for every 1 unit increase in \( x \), \( y \) decreases by \(\frac{2}{3}\) units.
2. **Quickly Plotting Points**: Since both equations are in slope-intercept form, plotting the lines can be done quickly:
- Start by plotting the y-intercept of each line on the graph.
- Use the slope to find a second point. For the first line, from (0, 2), you can move up 3 units and right 1 unit to find the point (1, 5), for example. For the second line, from (0, -4), move down 2 units and right 3 units to find the point (3, -6).
- Draw the lines through the plotted points.
This explicit structure of the equations enables a straightforward approach to plotting and visualizing the system of equations immediately without additional manipulation.