Problem 66
Question
a. Rewrite the given equation in slope-intercept form. b. Give the slope and \(y\) -intercept. c. Use the slope and y-intercept to graph the linear function. $$4 y+28=0$$
Step-by-Step Solution
Verified Answer
The slope-intercept form is \(y = -7\), the slope is 0, and the y-intercept is -7.
1Step 1: Convert to slope-intercept form
Start by isolating \(y\) in the equation \(4y+28=0\). To do this, subtract 28 from both sides and then divide by 4 to get \(y = -7\).
2Step 2: Identify the slope and y-intercept
The equation \(y = -7\) corresponds to a horizontal line that crosses the y-axis at -7. Therefore, the slope (m) is 0 and the y-intercept (b) is -7.
3Step 3: Graph the linear function
The line representing the function is a horizontal line that crosses the y-axis at -7. The slope or steepness of the line is zero, indicating that the line is horizontal. The line can be drawn starting at the point (0,-7) and continuing horizontally in both directions.
Key Concepts
Understanding Linear FunctionsDefining the SlopeIdentifying the Y-InterceptGraphing Linear Equations
Understanding Linear Functions
A linear function is a mathematical expression where each input has a corresponding output point that forms a straight line when graphed. This is typically represented in the form of an equation, such as the slope-intercept form: \( y = mx + b \). Here, \( y \) represents the output or dependent variable, \( m \) is the slope or the rate of change, \( x \) is the input or independent variable, and \( b \) is the y-intercept. In essence, linear functions are used to model relationships with constant rates of change.
For the equation \( y = -7 \) given in the exercise, this function indicates a constant value of \( y \) which implies a straightforward relationship. Because there’s no \( x \) term, it means the rate of change is zero, so you get a horizontal line on the graph.
For the equation \( y = -7 \) given in the exercise, this function indicates a constant value of \( y \) which implies a straightforward relationship. Because there’s no \( x \) term, it means the rate of change is zero, so you get a horizontal line on the graph.
Defining the Slope
The slope is a crucial part of understanding linear functions. It measures how steep the line is and describes the rate of change between the variables. The slope \( m \) is calculated as the "rise" (change in \( y \)) over "run" (change in \( x \)). In the standard slope-intercept form \( y = mx + b \), the \( m \) stands for the slope.
A positive slope means that as \( x \) increases, \( y \) also increases, whereas a negative slope means that as \( x \) increases, \( y \) decreases.
In our example, \( y = -7 \), the slope is \( m = 0 \). A zero slope indicates a horizontal line, which means there is no vertical change as \( x \) increases—\( y \) remains constant.
A positive slope means that as \( x \) increases, \( y \) also increases, whereas a negative slope means that as \( x \) increases, \( y \) decreases.
In our example, \( y = -7 \), the slope is \( m = 0 \). A zero slope indicates a horizontal line, which means there is no vertical change as \( x \) increases—\( y \) remains constant.
Identifying the Y-Intercept
The y-intercept is another important element in evaluating linear functions. This is the point where the line crosses the y-axis on a graph. In the slope-intercept form \( y = mx + b \), \( b \) represents the y-intercept.
If a line crosses the y-axis at the point (0, \( b \)), this tells us the value of \( y \) when \( x \) is zero.
For the equation \( y = -7 \), the y-intercept \( b \) is \(-7\). This means the line crosses the y-axis at the point (0, -7). Understanding the y-intercept helps in quickly locating points on a graph and can be handy when graphing or analyzing lines.
If a line crosses the y-axis at the point (0, \( b \)), this tells us the value of \( y \) when \( x \) is zero.
For the equation \( y = -7 \), the y-intercept \( b \) is \(-7\). This means the line crosses the y-axis at the point (0, -7). Understanding the y-intercept helps in quickly locating points on a graph and can be handy when graphing or analyzing lines.
Graphing Linear Equations
Graphing linear equations helps visualize the relationship between two variables. You often use the slope and y-intercept to start. Once you understand these aspects of a line, plotting it becomes simpler. First, mark the y-intercept on the graph, which is the point (0, \( b \)).
In summary, understanding how to graph a linear equation involves starting with the y-intercept and using the slope to determine the line's direction. This process visually represents the equation and helps comprehend the relationship between variables.
- From this point, use the slope (rise over run) to find additional points on the line.
- A positive slope means moving up and to the right, while a negative slope means moving down and to the right.
In summary, understanding how to graph a linear equation involves starting with the y-intercept and using the slope to determine the line's direction. This process visually represents the equation and helps comprehend the relationship between variables.
Other exercises in this chapter
Problem 66
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graph both equations in the same rectangular coordinate system and find all points of intersection. Then show that these ordered pairs satisfy the equations. $$
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