Problem 43
Question
Graph each pair of lines in the same coordinate system using the slope and y-intercept. $$ \begin{aligned} &y=\frac{2}{3} x-1\\\ &y=\frac{2}{3} x+1 \end{aligned} $$
Step-by-Step Solution
Verified Answer
Graph the lines by plotting points at (0, -1) and (3, 1) for the first line, and (0, 1) and (3, 3) for the second line.
1Step 1: Identify the slope and y-intercept of the first line
For the equation \(y = \frac{2}{3}x - 1\), the slope is \(\frac{2}{3}\) and the y-intercept is \(-1\).
2Step 2: Plot the y-intercept of the first line
On the graph, locate the point where the first line crosses the y-axis, which is at \(y = -1\). This point is \((0, -1)\).
3Step 3: Use the slope to plot another point for the first line
Starting at the y-intercept \((0, -1)\), use the slope \(\frac{2}{3}\) to find another point. From \((0, -1)\), move up 2 units and to the right 3 units to reach the point \((3, 1)\).
4Step 4: Draw the first line
Draw a straight line through the points \((0, -1)\) and \((3, 1)\). This is the graph of the first line.
5Step 5: Identify the slope and y-intercept of the second line
For the equation \(y = \frac{2}{3}x + 1\), the slope is \(\frac{2}{3}\) and the y-intercept is \(1\).
6Step 6: Plot the y-intercept of the second line
On the graph, locate the point where the second line crosses the y-axis, which is at \(y = 1\). This point is \((0, 1)\).
7Step 7: Use the slope to plot another point for the second line
Starting at the y-intercept \((0, 1)\), use the slope \(\frac{2}{3}\) to find another point. From \((0, 1)\), move up 2 units and to the right 3 units to reach the point \((3, 3)\).
8Step 8: Draw the second line
Draw a straight line through the points \((0, 1)\) and \((3, 3)\). This is the graph of the second line.
Key Concepts
SlopeY-InterceptCoordinate System
Slope
When working with linear equations, understanding the concept of slope is crucial. The slope of a line determines how steep the line is and the direction it leans.
Mathematically, the slope is represented by the letter \(m\). It is the ratio of the change in the y-values (vertical change) to the change in the x-values (horizontal change) between two points on the line.
The formula to calculate the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
For example, in the equation \(y = \frac{2}{3}x - 1\), the slope \(m\) is \(\frac{2}{3}\). This indicates that for every 3 units moved horizontally, the line rises by 2 units vertically.
Positive slopes incline upwards while negative slopes decline downwards.
Mathematically, the slope is represented by the letter \(m\). It is the ratio of the change in the y-values (vertical change) to the change in the x-values (horizontal change) between two points on the line.
The formula to calculate the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
For example, in the equation \(y = \frac{2}{3}x - 1\), the slope \(m\) is \(\frac{2}{3}\). This indicates that for every 3 units moved horizontally, the line rises by 2 units vertically.
Positive slopes incline upwards while negative slopes decline downwards.
- \(m = 0\) produces a horizontal line.
- Undefined slope (when dividing by 0) produces a vertical line.
Y-Intercept
The y-intercept is a point where the line crosses the y-axis. In the equation of a line \(y = mx + b\), the y-intercept is represented by the letter \(b\).
This value indicates the point where the line meets the y-axis when \(x = 0\).
For example, in the equation \(y = \frac{2}{3}x - 1\), the y-intercept \(b\) is \(-1\). This tells us that the line crosses the y-axis at the point \((0, -1)\).
Knowing the y-intercept helps in plotting the line on a graph.
Here are key points to remember:
This value indicates the point where the line meets the y-axis when \(x = 0\).
For example, in the equation \(y = \frac{2}{3}x - 1\), the y-intercept \(b\) is \(-1\). This tells us that the line crosses the y-axis at the point \((0, -1)\).
Knowing the y-intercept helps in plotting the line on a graph.
Here are key points to remember:
- The y-intercept gives you an initial point to start drawing the line.
- The value of \(b\) can be either positive or negative depending on where the line crosses the y-axis.
Coordinate System
The coordinate system is a framework used to graph equations like lines. It consists of two perpendicular axes called the x-axis (horizontal) and the y-axis (vertical).
The point where these axes intersect is called the origin, denoted as \((0, 0)\).
Each point on the graph is identified by a pair of coordinates \((x, y)\). The x-value represents the horizontal position, and the y-value represents the vertical position.
For example, the point \((3, 2)\) is 3 units to the right of the origin and 2 units up.
Here are a few key points to keep in mind:
The point where these axes intersect is called the origin, denoted as \((0, 0)\).
Each point on the graph is identified by a pair of coordinates \((x, y)\). The x-value represents the horizontal position, and the y-value represents the vertical position.
For example, the point \((3, 2)\) is 3 units to the right of the origin and 2 units up.
Here are a few key points to keep in mind:
- Each point on the graph has a unique set of coordinates.
- The coordinate system allows for visual representation of algebraic equations.
- Lines are graphed by plotting points and then drawing straight lines through these points.
Other exercises in this chapter
Problem 42
Graph each pair of lines in the same coordinate system using the slope and y-intercept. $$ \begin{aligned} &y=2 x+3\\\ &y=-\frac{1}{2} x+3 \end{aligned} $$
View solution Problem 42
Graph each linear equation. Plot four points for each line. $$x+2 y=4$$
View solution Problem 43
Graph each linear equation. Plot four points for each line. $$6 x+3 y=0$$
View solution Problem 44
Graph each pair of lines in the same coordinate system using the slope and y-intercept. $$ \begin{aligned} &y=-2 x+4\\\ &y=-2 x+2 \end{aligned} $$
View solution