Problem 32 For each equation, (a) write it ... [FREE SOLUTION]

Chapter 3: Problem 32

For each equation, (a) write it in slope-intercept form, (b) give the slope of the line, (c) give the y-intercept, and (d) graph the line. $$ 7 x-3 y=3 $$

Short Answer

Expert verified

Slope-intercept form: $ y = \frac{7}{3}x - 1 $, Slope: $ \frac{7}{3} $, y-intercept: -1. Graph by plotting (0, -1) and (3, 6).

Step by step solution

Write the equation in slope-intercept form

The slope-intercept form of a line is given by the equation $ y = mx + b $, where $ m $ is the slope and $ b $ is the y-intercept. Start with the given equation: $ 7x - 3y = 3 $. To write it in slope-intercept form, solve for $ y $.First, isolate $ y $ by moving the term involving $ x $ to the right side: $ -3y = -7x + 3 $. Then, divide each term by $ -3 $: $ y = \frac{7}{3}x - 1 $. The equation in slope-intercept form is $ y = \frac{7}{3}x - 1 $.

Identify the slope

The slope $ m $ is the coefficient of $ x $ in the slope-intercept form $ y = mx + b $. From the equation $ y = \frac{7}{3}x - 1 $, the slope is $ m = \frac{7}{3} $.

Identify the y-intercept

The y-intercept $ b $ is the constant term in the slope-intercept form $ y = mx + b $. From the equation $ y = \frac{7}{3}x - 1 $, the y-intercept is $ b = -1 $.

Graph the line

To graph the line, start by plotting the y-intercept $ b $. Here, the y-intercept is $ -1 $, so plot the point (0, -1) on the y-axis. Then, use the slope $ m = \frac{7}{3} $ to find another point. The slope means rise over run, so from the y-intercept, move up 7 units and right 3 units to arrive at the point (3, 6). Plot this point as well. Draw a line through the points (0, -1) and (3, 6) to graph the line.

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Key Concepts

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

Solving Linear Equations

Understanding how to solve linear equations is foundational for mastering algebra. A linear equation is any equation that can be written in the form ax + by = c. The goal is usually to isolate the variable y and write the equation in a form that is easy to interpret. For example, consider the equation given: $7x - 3y = 3$. To solve for $y$, you need to rearrange the terms. First, move the $x$ term to the other side of the equation, resulting in $-3y = -7x + 3$. Next, divide each term by $-3$ to isolate $y$, which gives $y = \frac{7}{3}x - 1$. This format, $y = mx + b$, is known as slope-intercept form and is crucial for graphing lines.

Slope of a Line

The slope of a line tells you how steep the line is. It's a measure of how much $y$ increases or decreases as $x$ increases. The slope is represented by $m$ in the slope-intercept form $y = mx + b$. Let's look at our equation $y = \frac{7}{3}x - 1$. Here, the slope $m$ is $\frac{7}{3}$. This means that for every 3 units you move horizontally to the right (run), you move 7 units vertically up (rise). The slope is essentially the ratio of the vertical change to the horizontal change between any two points on the line.

Y-Intercept

The y-intercept is the point where the line crosses the y-axis. In the slope-intercept form $y = mx + b$, the y-intercept is represented by $b$. It gives us a starting point for graphing the line. From our example equation $y = \frac{7}{3}x - 1$, we see that the y-intercept $b$ is $-1$. This tells us that the line crosses the y-axis at the point (0, -1). To graph this, you simply plot the point (0, -1), which serves as our initial point before using the slope to find other points on the line.

Graphing Linear Equations

Once you've identified the slope and y-intercept, graphing the linear equation becomes straightforward. Start by plotting the y-intercept, the point where the line crosses the y-axis. For our example $y = \frac{7}{3}x - 1$, plot the point (0, -1). Next, use the slope to find another point. Since the slope is $\frac{7}{3}$, starting from the y-intercept (0, -1), move up 7 units and to the right 3 units. This brings you to the point (3, 6). Placing both points on the graph, draw a straight line through them. This line represents the equation $y = \frac{7}{3}x - 1$, showing visually how y depends on x.

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91影视

For each equation, (a) write it in slope-intercept form, (b) give the slope of the line, (c) give the y-intercept, and (d) graph the line. $$ 7 x-3 y=3 $$

Short Answer

Step by step solution

Write the equation in slope-intercept form

Identify the slope

Identify the y-intercept

Graph the line

Key Concepts

Solving Linear Equations

Slope of a Line

Y-Intercept

Graphing Linear Equations

One App. One Place for Learning.

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