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Chapter 2: Problem 33

Determine the slope and the \(y\) -intercept. $$ y=\frac{2}{3} x+4 $$

Short Answer

Expert verified
The slope is \( \frac{2}{3} \) and the y-intercept is \( 4 \).

Step by step solution

01

- Identify the slope-intercept form

The given equation, \( y = \frac{2}{3}x + 4 \), is in slope-intercept form, which is \( y = mx + b \). In this form, \( m \) is the slope, and \( b \) is the y-intercept.
02

- Determine the slope

From the equation, compare it with \( y = mx + b \) to find that \( m = \frac{2}{3} \). Therefore, the slope is \( \frac{2}{3} \).
03

- Determine the y-intercept

In the equation \( y = \frac{2}{3}x + 4 \), the constant term \( 4 \) is the y-intercept \( b \). Thus, the y-intercept is \( 4 \).

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Key Concepts

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

slope-intercept form
The slope-intercept form of a linear equation is one of the most common ways to express linear equations. It's written as \( y = mx + b \). This form is particularly useful because it makes it easy to identify the slope and the y-intercept of a line. The variable \( m \) represents the slope, which indicates the steepness or incline of the line. It tells you how much \( y \) changes for a unit change in \( x \). The variable \( b \) stands for the y-intercept, the point where the line crosses the y-axis. Being familiar with the slope-intercept form helps you quickly graph linear equations and understand the relationship between the variables.
linear equations
Linear equations are equations of the first degree, which means they have no exponents greater than one. They are called 'linear' because their graph is a straight line. Linear equations can be written in multiple forms, but the slope-intercept form is one of the most intuitive. These equations are fundamental in algebra because they allow us to model and solve real-world problems by finding the relationship between two variables. When you know the slope and y-intercept, you can easily draw or visualize the line that represents the equation.
algebra
Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. It's a unifying thread of almost all of mathematics and includes everything from solving simple equations to studying abstractions like groups, rings, and fields. In the context of the problem, algebra helps us understand the relationship between the variables \( x \) and \( y \) through their linear equation. By applying algebraic principles, we can transform and interpret equations to uncover meaningful insights, such as the slope and y-intercept of a line. Understanding algebra is crucial for tackling more complex mathematics and real-world applications.

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Most popular questions from this chapter

Explain the difference between finding \(f(0)\) and finding the input \(x\) for which \(f(x)=0\)

For equation, select a window that shows the curvature of the graph and create a table of ordered pairs in which \(x\) -values extend, by tenths, from 0 to 0.6. a) \(y=0.375 x^{3}\) b) \(y=-3.5 x^{2}+6 x-8\) c) \(y=(x-3.4)^{3}+5.6\)

Solve each equation graphically. Then check your answer by solving the same equation algebraically. $$ x+4=3 x+5 $$

Without graphing, how can you tell that the graph of \(y=x-30\) passes through three quadrants?

Assume that \(r, p,\) and \(s\) are nonzero constants and that \(x\) and \(y\) are variables. Determine whether each equation is linear. $$ p y=s x-r^{2} y-9 $$
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