Problem 45
Question
Give the slope and y-intercept of each line whose equation is given. Then graph the line. $$y=-\frac{3}{5} x+7$$
Step-by-Step Solution
Verified Answer
The slope of the line is \(-\frac{3}{5}\) and the y-intercept is \(7\). The line crosses the y-axis at \(7\) and the slope guides it to descend as we proceed in the positive direction of the x-axis.
1Step 1: Identify the Slope and the Y-intercept
From the given linear equation \(y=-\frac{3}{5}x+7\), it's seen that the coefficient of \(x\) which is \( -\frac{3}{5}\) is the slope (\(m\)), and the constant term \(7\) is the y-intercept (\(c\)).
2Step 2: Plot the Y-intercept
Start with a graph. Plot the y-intercept on the y-axis. So, draw a dot on \(y = 7\).
3Step 3: Plot the Slope
The slope of a line is calculated as rise over run. Here, it means move three units downward (\(-3\)) and five units to the right (\(5\)). From the y-intercept, we can proceed with these measurements and draw a dot to represent the slope.
4Step 4: Draw the Line
Draw a straight line that goes through these two points (the Y-intercept and the point obtained from slope). This is the graphical representation of the equation.
Key Concepts
SlopeY-interceptGraphing LinesAlgebra
Slope
When we talk about the slope in the context of linear equations, we are referring to how steep the line is. In a linear equation of the form \(y = mx + c\), the slope is denoted by \(m\). It tells us how much the \(y\) value, or vertical change, increases or decreases for a unit change in \(x\).
For the equation \(y = -\frac{3}{5}x + 7\), the slope is \(-\frac{3}{5}\). This means for every 5 units you move to the right along the x-axis, the line moves down 3 units along the y-axis.
To summarize the main points about slope:
For the equation \(y = -\frac{3}{5}x + 7\), the slope is \(-\frac{3}{5}\). This means for every 5 units you move to the right along the x-axis, the line moves down 3 units along the y-axis.
To summarize the main points about slope:
- It is represented as a ratio: the rise over the run.
- It signifies the direction and steepness of a line.
- A positive slope means the line rises from left to right, while a negative slope means it falls.
- A zero slope indicates a horizontal line, and an undefined slope indicates a vertical line.
Y-intercept
The y-intercept is another crucial component of the linear equation. It's the point where the line crosses the y-axis. In our equation \(y = -\frac{3}{5}x + 7\), the y-intercept is the constant term \(7\).
This tells us that when \(x\) is zero, \(y\) will be equal to 7.
Understanding the y-intercept allows us to easily plot the starting point on a graph.
Here are some key points:
This tells us that when \(x\) is zero, \(y\) will be equal to 7.
Understanding the y-intercept allows us to easily plot the starting point on a graph.
Here are some key points:
- The y-intercept provides a fixed starting point on the graph, making it easier to draw the line accurately.
- It is found in every linear equation in the format \(c\) in \(y = mx + c\).
- When graphing, this is the point at which you begin before using the slope to draw the rest of the line.
Graphing Lines
Graphing a line requires understanding both the slope and the y-intercept. First, plot the y-intercept on the graph at \(y = 7\).
Next, use the slope \(-\frac{3}{5}\) to locate the next point. This means from the y-intercept, you move down 3 units and 5 units to the right to find a second point on the line.
Once you have these two points, you simply draw a straight line through them.
Some tips for graphing lines:
Next, use the slope \(-\frac{3}{5}\) to locate the next point. This means from the y-intercept, you move down 3 units and 5 units to the right to find a second point on the line.
Once you have these two points, you simply draw a straight line through them.
Some tips for graphing lines:
- Verify your calculations, especially when finding additional points using the slope from the y-intercept.
- Always label your axes to avoid confusion.
- Extend the line in both directions to cover the graph as much as possible.
- Make sure your graph is neat, which helps when cross-checking your work.
Algebra
Algebra is the tool we use to decipher and handle linear equations like \(y = -\frac{3}{5}x + 7\). It allows us to understand relationships between variables and how they change. With linear equations, you're looking for a relationship that can be plotted as a straight line.
Key algebra concepts related to linear equations include solving for variables and manipulating equations to reveal slope and y-intercept.
Important things to remember about algebra in this context:
Key algebra concepts related to linear equations include solving for variables and manipulating equations to reveal slope and y-intercept.
Important things to remember about algebra in this context:
- Algebra provides techniques to rearrange equations into different useful forms.
- Understanding algebraic manipulation can make it easier to solve complex problems with multiple steps.
- It's fundamental for finding the slope and y-intercept and, thus, graphing the line effectively.
- Proper utilization of algebra can quickly reveal important characteristics of the graph.
Other exercises in this chapter
Problem 44
In Exercises \(33-44\), find and simplify the difference quotient $$\frac{f(x+h)-f(x)}{h}, h \neq 0$$for the given function. $$f(x)=\frac{1}{2 x}$$
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Give the center and radius of the circle described by the equation and graph each equation. $$ (x-2)^{2}+(y-3)^{2}=16 $$
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Evaluate each piecewise function at the given values of the independent variable. $$\begin{aligned} &f(x)=\left\\{\begin{array}{ll} 3 x+5 & \text { if } x
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Express the given function h as a composition of two functions f and g so that \(h(x)=(f \circ g)(x)\) $$h(x)=\frac{1}{2 x-3}$$
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