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Chapter 3: Problem 63

(a) find the y-intercept. (b) find the x-intercept. (c) find a third solution of the equation. (d) graph the equation. \(2 x+3 y=4\)

Short Answer

Expert verified
y-intercept: (0, \(\frac{4}{3}\)); x-intercept: (2, 0); third solution: (1, \(\frac{2}{3}\)); graph the line through these points.

Step by step solution

01

- Find the y-intercept

To find the y-intercept, set x to 0 and solve for y. Start with the equation: \(2x + 3y = 4\)Set x to 0:\(2(0) + 3y = 4\)Which simplifies to:\(3y = 4\)Finally, solve for y:\(y = \frac{4}{3}\)So, the y-intercept is (0, \(\frac{4}{3}\)).
02

- Find the x-intercept

To find the x-intercept, set y to 0 and solve for x. Start with the equation: \(2x + 3y = 4\)Set y to 0:\(2x + 3(0) = 4\)Which simplifies to:\(2x = 4\)Finally, solve for x:\(x = 2\)So, the x-intercept is (2, 0).
03

- Find a third solution

Choose a value for x or y and solve for the other variable. Let's choose x = 1.Start with the equation: \(2x + 3y = 4\)Set x to 1: \(2(1) + 3y = 4\)Which simplifies to:\(2 + 3y = 4\)Subtract 2 from both sides:\(3y = 2\)Finally, solve for y:\(y = \frac{2}{3}\)So, the third solution is (1, \(\frac{2}{3}\)).
04

- Graph the equation

Plot the intercepts and the third solution on the coordinate plane: 1. Plot the y-intercept (0, \(\frac{4}{3}\)).2. Plot the x-intercept (2, 0).3. Plot the third solution (1, \(\frac{2}{3}\)).Draw a straight line through all three points to graph the equation.

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Key Concepts

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

y-intercept
When working with linear equations, the y-intercept is the point where the line crosses the y-axis. This happens when the value of x is 0. To find it, you substitute x with 0 in the equation and solve for y. For the equation given, which is: \(2x + 3y = 4\) set x to 0 and it becomes \(3y = 4\). Solving for y gets you \(y = \frac{4}{3}\). So, the y-intercept is (0, \(\frac{4}{3}\)). Whenever you see 'intercept', think about zeroing out the other variable!
x-intercept
The x-intercept is the point where the line crosses the x-axis. This happens when the value of y is 0. To find it, substitute y with 0 in the equation and solve for x. With the same equation: \(2x + 3y = 4\), set y to 0 and it simplifies to \(2x = 4\). Solving for x gets you \(x = 2\). Thus, the x-intercept is (2, 0). Remember, to find the x-intercept, make y zero!
graphing linear equations
Graphing linear equations involves plotting points and drawing a line through them. First, you plot the intercepts (where the line crosses the x-axis and y-axis). In our example, these intercepts are (0, \(\frac{4}{3}\)) and (2, 0). You may also need another point for accuracy. For this, you choose another value for x or y, plug it into the equation, and see where it falls. In our example, choosing x = 1, the calculation shows another point at (1, \(\frac{2}{3}\)). Once you have these points, you draw a straight line through them to represent the equation on a graph.
solving linear equations
Solving linear equations is about finding values for variables that make the equation true. Generally, you start by isolating one of the variables. For instance, given: \(2x + 3y = 4\)You can find the intercepts by setting x or y to zero and solving for the other variable. Once you identify these points, you can use one of them to check additional values. These steps help you confirm that the relationship described by the equation holds true. Always remember, small mistakes can trip you up, so take it step by step!

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Most popular questions from this chapter

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 strategies listed that you don't currently use but you think might be helpful.

For exercises 1-8, (a) represent the information as two ordered pairs. (b) find the average rate of change, \(m\). The amount of certified organic cropland in Washington State planted in peas increased from 28 acres in 2007 to 252 acres in 2010. Round to the nearest whole number. (Source: www.tfrec.wsu.edu, March 2011)

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

(a) represent the information as two ordered pairs. (b) find the average rate of change, \(m\). The amount of fresh tomatoes consumed per person in the United States increased from \(89.9\) lb in 2009 to 93.5 lb in 2011. (Source: www.ers.usda.gov, Dec. 15,2011\()\)

(a) represent the information as two ordered pairs. (b) find the average rate of change, \(m\). The number of women enrolled in the fall in degreegranting institutions of higher education increased from \(10,184,000\) women in 2006 to \(11,658,000\) women in 2009. Round to the nearest thousand. (Source: nces .ed.gov, 2011)
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