Problem 20 Construct an equation and sketch... [FREE SOLUTION]

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Explorations in College Algebra

Linda Almgren Kime, Judy Clark, Beverly K. Michael

$Math Studyset 91影视 Explanations$ Math

4 Edition

Chapter 2: Problem 20

Construct an equation and sketch the graph of its line with the given slope, $m,$ and vertical intercept, $b .$ (Hint: Find two points on the line.) a. $m=2, b=-3$ b. $m=-\frac{3}{4}, b=1$ c. $m=0, b=50$

Short Answer

Expert verified

a.$ y = 2x - 3 $; b.$ y = -\frac{3}{4}x + 1 $; c.$ y = 50 $.

Step by step solution

- Understand the Equation of a Line

The general form of a linear equation is given by: \[ y = mx + b \]where $ m $ is the slope and $ b $ is the y-intercept. Given the values of $ m $ and $ b $, we can construct the equation of the line.

- Construct the Equation for Part (a)

Given: $ m = 2 $ and $ b = -3 $. Substitute these values into the general form: \[ y = 2x - 3 \]

- Find Two Points for Part (a)

To sketch the line, find two points. Let's choose two values for x and compute y.For $ x = 0 $: \[ y = 2(0) - 3 \]\[ y = -3 \]So, the point is (0, -3).For $ x = 1 $: \[ y = 2(1) - 3 \]\[ y = -1 \]So, the point is (1, -1).

- Sketch the Graph for Part (a)

Plot the points (0, -3) and (1, -1) on a graph and draw a line passing through them. This line represents the equation $ y = 2x - 3 $.

- Construct the Equation for Part (b)

Given: $ m = -\frac{3}{4} $ and $ b = 1 $. Substitute these values into the general form: \[ y = -\frac{3}{4}x + 1 \]

- Find Two Points for Part (b)

To sketch the line, find two points. Let's choose two values for x and compute y.For $ x = 0 $: \[ y = -\frac{3}{4}(0) + 1 \]\[ y = 1 \]So, the point is (0, 1).For $ x = 4 $: \[ y = -\frac{3}{4}(4) + 1 \]\[ y = -3 + 1 \]\[ y = -2 \]So, the point is (4, -2).

- Sketch the Graph for Part (b)

Plot the points (0, 1) and (4, -2) on a graph and draw a line passing through them. This line represents the equation $ y = -\frac{3}{4}x + 1 $.

- Construct the Equation for Part (c)

Given: $ m = 0 $ and $ b = 50 $. Substitute these values into the general form: \[ y = 0x + 50 \]\[ y = 50 \]

- Find Two Points for Part (c)

To sketch the line, find two points. Let's choose two values for x (since y is constant).For $ x = 0 $: \[ y = 50 \]So, the point is (0, 50).For $ x = 1 $: \[ y = 50 \]So, the point is (1, 50).

- Sketch the Graph for Part (c)

Plot the points (0, 50) and (1, 50) on a graph and draw a horizontal line passing through them. This line represents the equation $ y = 50 $.

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Key Concepts

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

linear equations

Linear equations are fundamental in algebra. They describe a straight line in the coordinate system. The general form of a linear equation is given by $ y = mx + b $. Here, $ y $ represents the dependent variable, $ x $ represents the independent variable, $ m $ is the slope of the line, and $ b $ is the y-intercept.
The slope determines the steepness and the direction of the line, while the y-intercept is the point where the line crosses the y-axis. Every linear equation will graph as a straight line, and understanding how to form and graph these equations is a crucial skill in mathematics.

slope-intercept form

The slope-intercept form of a linear equation is one of the most commonly used forms, expressed as $ y = mx + b $.
Here, $ m $ represents the slope of the line, and $ b $ is the y-intercept. This form is very useful because it provides immediate information about the slope and the y-intercept, making it easy to graph the line.
In the equation $ y = 2x - 3 $, for example, the slope $ m $ is 2, meaning the line rises 2 units for every 1 unit it moves to the right. The y-intercept $ b $ is -3, which tells us the line crosses the y-axis at -3. This concise format makes it simple to construct and understand the behavior of the line.

graphing lines

Graphing a line involves plotting points and drawing a straight line through them. Begin by identifying the slope and y-intercept from the equation in the slope-intercept form.
For example, given the equation $ y = 2x - 3 $, the slope is 2 and the y-intercept is -3.
Start by plotting the y-intercept on the graph (0, -3). This is where the line crosses the y-axis. Next, use the slope to find another point. A slope of 2 means that for every 1 unit you move to the right on the x-axis, you move up 2 units on the y-axis.
Plot this second point, which in this case would be (1, -1). Finally, draw a straight line through these points. Repeat the process for other equations to visualize different lines on the graph.

y-intercept

The y-intercept is a key component of a linear equation. It is the point where the line crosses the y-axis, and it is represented by the $ b $ value in the slope-intercept form $ y = mx + b $.
For example, in the equation $ y = -\frac{3}{4}x + 1 $, the y-intercept is 1, meaning the line crosses the y-axis at (0, 1). The y-intercept provides a starting point for graphing the line.
By understanding the y-intercept, you can quickly determine where the line begins in the y-dimension, making it easier to graph the equation accurately and understand its behavior.

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91影视

Construct an equation and sketch the graph of its line with the given slope, \(m,\) and vertical intercept, \(b .\) (Hint: Find two points on the line.) a. \(m=2, b=-3\) b. \(m=-\frac{3}{4}, b=1\) c. \(m=0, b=50\)