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Chapter 3: Problem 84

Use slope-intercept graphing to graph the equation. $$ y=-\frac{2}{5} x+4 $$

Short Answer

Expert verified
Slope: \(-\frac{2}{5}\), y-intercept: 4, points (0, 4) and (5, 2).

Step by step solution

01

- Identify the slope and y-intercept

The given equation is in slope-intercept form, which is \[ y = mx + b \]Here, the slope \( m \) is \(-\frac{2}{5}\) and the y-intercept \( b \) is 4.
02

- Plot the y-intercept

To plot the y-intercept, find point (0, 4) on the y-axis and place a dot. This is the point where the line crosses the y-axis.
03

- Use the slope to find another point

The slope \(-\frac{2}{5}\) tells us to move down 2 units and right 5 units from the y-intercept. Starting at (0, 4), move to the right 5 units to (5, 4). Then, move down 2 units to (5, 2). Plot this point (5, 2).
04

- Draw the line

Using a ruler, draw a straight line through the points (0, 4) and (5, 2). This line represents the graph of the given equation.

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Key Concepts

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

graphing
Graphing linear equations can help visualize the relationship between variables. It's a way to see how changing the value of one variable affects the other. When we graph an equation, we turn it into a picture, making it easier to interpret and understand.

By plotting points and drawing lines, we see the equation in action. This visual representation can be particularly helpful when solving problems that involve variables. It also aids in identifying patterns and trends within the data.
linear equations
A linear equation is an equation between two variables that gives a straight line when plotted on a graph. The most common form of a linear equation is the slope-intercept form, which is written as \[ y = mx + b \].

In this equation, \(m\) represents the slope, which shows the steepness of the line, and \(b\) represents the y-intercept, which is where the line crosses the y-axis. Linear equations are fundamental in algebra and are used in various real-life situations to model relationships between quantities.
algebra basics
Understanding the basics of algebra is crucial for mastering more complex topics. In algebra, we deal with numbers and variables, and we use letters to represent unknown values.

One of the foundational concepts in algebra is solving equations. This involves finding the value of variables that make the equation true. Another key concept is graphing, where we plot equations on a coordinate plane to visually represent relationships between variables. Building a strong foundation in these basics makes tackling advanced problems much easier.
slope and y-intercept
The slope and y-intercept are two key components in the slope-intercept form of a linear equation. The slope, usually represented by the letter \(m\), indicates the rate of change between the variables. It's the 'rise over run' value, showing how much \(y\) changes for a given change in \(x\).

The y-intercept, denoted by \(b\), is the point where the line crosses the y-axis. This intercept is important because it provides a starting point for graphing the equation. Together, the slope and y-intercept provide complete information to accurately graph a line.

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Most popular questions from this chapter

(a) find the \(y\)-intercept. (b) find the \(x\)-intercept. (c) use the slope formula to find the slope of the line. \(2 x+5 y=15\)

A high-speed Shinkansen train in Japan travels at a speed of \(\frac{270 \mathrm{~km}}{1 \mathrm{hr}}\) for \(18 \mathrm{~min}\). Find the distance it travels.

For exercises 97-98, some students find it helpful to use their learning preferences as a guide in how to study. Visual Learner \- Take detailed notes during class. Use colored pens and highlighters. \- Reorganize and rewrite notes after class; draw diagrams that summarize what you have learned. \- Read your book; watch the videos or DVDs for this text. \- Use flash cards for memory work. \- Sit where you can see everything in the classroom. Turn your phone or tablet off so that you are not distracted. Auditory Learner \- With permission, record your class. Take only brief notes of the big ideas and examples. After class, listen to the recording. Complete your notes. Restate the main ideas aloud to yourself. Use videos and DVDs to fill in anything you missed in class. \- Talk to yourself as you do your homework. Explain each step to yourself. \- Do memory work by repeating definitions aloud. Listen to a recording of the words and definitions. Create songs that help you remember a definition. \- Sit where you can hear everything. Turn your phone or tablet off so that you are not distracted. Kinesthetic Learner \- With permission, record your class. Take brief notes of the big ideas and examples. After class, listen to the recording. Complete your notes. Draw pictures. Use videos and DVDs to fill in anything you missed during class. -With your finger, trace diagrams and graphs. Do not just look at them. \- Imagine symbols such as variables as three-dimensional objects or even cartoon characters. Imagine yourself counting them, combining them, or subtracting them. \- Do memory work as you exercise or walk to your car. Walk around your room as you repeat definitions. You may find it helpful to come up with physical motions and/or a song that correspond to a procedure. \- If your class is mostly lecture, prepare yourself mentally before you walk into class to concentrate and not daydream. Turn your phone or tablet off so that you are not distracted. Identify any of the strategies listed above that you currently use to study math.

(a) write the equation of the vertical line that passes through the point. (b) graph the equation. \((-2,-1)\)

(a) write the equation of the horizontal line that passes through the point. (b) graph the equation. \((-3,-2)\)
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