/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 37 For each equation, identify the ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 37

For each equation, identify the slope and the y-intercept. Graph the line to check your answer. $$ y=\frac{3}{4}+\frac{1}{2} x $$

Short Answer

Expert verified
Slope: \( \frac{1}{2} \), Y-intercept: \( \frac{3}{4} \).

Step by step solution

01

Rewrite the Equation

Rewrite the given equation in the slope-intercept form, which is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.The given equation is \(y = \frac{3}{4} + \frac{1}{2}x\). Rearrange to match the slope-intercept form:\(y = \frac{1}{2}x + \frac{3}{4}\).
02

Identify the Slope

In the slope-intercept form \(y = mx + b\), \(m\) represents the slope.From the equation \(y = \frac{1}{2}x + \frac{3}{4}\), the slope \(m\) is \(\frac{1}{2}\).
03

Identify the Y-Intercept

In the slope-intercept form \(y = mx + b\), \(b\) represents the y-intercept.From the equation \(y = \frac{1}{2}x + \frac{3}{4}\), the y-intercept \(b\) is \(\frac{3}{4}\).
04

Graph the Line

Plot the y-intercept \( (0, \frac{3}{4}) \) on the graph. From this point, use the slope \( \frac{1}{2} \) to find another point. Since the slope tells us to go up 1 unit and right 2 units (rise/run), plot the next point at \( (2, \frac{5}{4}) \). Draw a line through these points to graph the line.
05

Verify Your Answer

Check the graph to ensure that the line correctly reflects the slope \(\frac{1}{2}\) and y-intercept \(\frac{3}{4}\). The plotted points should align, confirming that the slope and y-intercept were identified accurately.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

linear equations
Linear equations are algebraic expressions representing a straight line when graphed on a coordinate plane. These equations are typically in the form of:
  • Ax + By = C (standard form)
  • y = mx + b (slope-intercept form)
In each form, the variables x and y stand for coordinates on the plane. In slope-intercept form, 'm' represents the slope of the line, and 'b' represents the y-intercept, which is where the line crosses the y-axis.
graphing linear functions
Graphing linear functions involves plotting points on a coordinate plane to represent the equation visually. Here is how you can do it step-by-step:
1. Convert the equation to slope-intercept form, i.e., y = mx + b.
2. Identify the y-intercept (b). This is the point where the line crosses the y-axis, and it has coordinates (0, b).
3. Plot the y-intercept on the graph.
4. Use the slope (m), which tells us how steep the line is, to determine the next point. The slope is the ratio of the rise (change in y) over the run (change in x). If m is 陆, for example, from the y-intercept, move up 1 unit and right 2 units to place the next point.
5. Draw a straight line through both points to extend the line across the graph.
By following these steps, you can graph any linear equation accurately.
slope-intercept form
The slope-intercept form of a linear equation is written as y = mx + b. This format is very useful because it directly reveals two key pieces of information:
  • The slope (m): This determines the angle and direction of the line. A positive slope means the line goes upward as it moves from left to right, while a negative slope means it goes downward.

  • The y-intercept (b): This is where the line crosses the y-axis. It's the value of y when x is 0.


For example, in the equation y = 陆x + 戮, the slope (m) is 陆, and the y-intercept (b) is 戮. This means from the y-intercept, the line rises 1 unit for every 2 units it runs to the right. This form makes it straightforward to both graph the equation and understand the nature of the line it represents.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

You have studied formulas to calculate the area and perimeter of various shapes. Some of these formulas are linear, and some aren鈥檛. Tell whether the formula for each measurement below is linear or not, and explain your answer. a. area of a circle b. circumference of a circle c. area of a square d. perimeter of a square

Below is world population data for the years 1950 through 1990. $$\begin{array}{|c|c|}\hline & {\text { Population }} \\ {\text { Year }} & {\text { (billions) }} \\ \hline 1950 & {2.52} \\ {1960} & {3.02} \\ {1970} & {3.70} \\ {1980} & {4.45} \\ {1990} & {5.29} \\ \hline\end{array}$$ a. Plot the points on a graph with 鈥淵ears since 1900鈥� on the horizontal axis and 鈥淧opulation (billions)鈥� on the vertical axis. Try to fit a line to the data. b. Write an equation to fit your line. c. Use your equation to project the world population for the year 2010, which is 110 years after 1900. d. What does your equation tell you about world population in 1900? Does this make sense? Explain. e. According to United Nations figures, the world population in 1900 was 1.65 billion. The UN has predicted that world population in the year 2010 will be 6.79 billion. Are the 1900 data and the prediction for 2010 different from your predictions? How do you explain your answer?

The lines for these three equations all pass through a common point. $$y=\frac{x}{2}-1 \quad y=-\frac{2 x}{3}+6 \quad y=-\frac{x}{6}+3$$ a. Draw graphs for the three equations, and find the common point. b. Verify that the point you found satisfies all three equations by substituting the x- and y-coordinates into each equation.

In this exercise, you will apply what you have learned about writing equations for parallel lines. a. Write three equations whose graphs are parallel lines with positive slopes. Write the equations so that the graphs are equally spaced. b. Graph the lines, and verify that they are parallel. c. Write three equations whose graphs are parallel lines with negative slopes and are equally spaced. d. Graph the lines, and verify that they are parallel.

For each equation, identify the slope and the y-intercept. Graph the line to check your answer. $$ y=-x+5 $$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.