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Chapter 2: Problem 45

Give the slope and y-intercept of each line whose equation is given. Then graph the line. $$y=-\frac{3}{5} x+7$$

Short Answer

Expert verified
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.

Step by step solution

01

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 (\(c\)).
02

Plot the Y-intercept

Start with a graph. Plot the y-intercept on the y-axis. So, draw a dot on \(y = 7\).
03

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.
04

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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:
  • 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:
  • 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:
  • 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:
  • 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.

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Most popular questions from this chapter

Describe one advantage of using \(f(x)\) rather than \(y\) in a function's equation. For people filing a single return, federal income tax is a function of adjusted gross income because for each value of adjusted gross income there is a specific tax to be paid. On the other hand, the price of a house is not a function of the lot size on which the house sits because houses on same-sized lots can sell for many different prices. a. Describe an everyday situation between variables that is a function. b. Describe an everyday situation between variables that is not a function.

Explain how to use the general form of a line's equation to find the line's slope and \(y\) -intercept.

Find a linear equation in slope-intercept form that models the given description. Describe what each variable in your model represents. Then use the model to make a prediction. In \(1995,60 \%\) of U.S. adults read a newspaper and this percentage has decreased at a rate of \(0.7 \%\) per year since then.

You will be developing functions that model given conditions. You commute to work a distance of 40 miles and return on the same route at the end of the day. Your average rate on the return trip is 30 miles per hour faster than your average rate on the outgoing trip. Write the total time, \(T,\) in hours, devoted to your outgoing and return trips as a function of your rate on the outgoing trip. Then find and interpret \(T(30) .\) Hint: Time traveled \(=\frac{\text { Distance traveled }}{\text { Rate of travel }}\)

You will be developing functions that model given conditions. A chemist working on a flu vaccine needs to mix a \(10 \%\) sodium-iodine solution with a \(60 \%\) sodium-iodine solution to obtain a 50 -milliliter mixture. Write the amount of sodium iodine in the mixture, \(S,\) in milliliters, as a function of the number of milliliters of the \(10 \%\) solution used. Then find and interpret \(S(30)\)
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