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Chapter 0: Problem 23

Find the slope and the \(y\) -intercept (if possible) of the line. $$x+5 y=20$$

Short Answer

Expert verified
The slope of the line is -1/5 and the y-intercept is 4.

Step by step solution

01

Convert to Slope-Intercept Form

The supplied linear equation is in standard form (Ax + By = C). The first step is to convert this to slope-intercept form, which is represented as y = mx + b. Let's start by rearranging the equation x + 5y = 20 as follows: \nSubtract x from both sides to get 5y = -x + 20.
02

Isolate y

Now, isolate y by dividing each term of the equation by 5. So, the equation becomes y = -1/5 * x + 4.
03

Identify the slope and the y-intercept

Now, we can simply read off the slope and the y-intercept from the equation. Comparing y = -1/5 * x + 4 with y = mx + b, we see that the slope (m) is -1/5 and the y-intercept (b) is 4.

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Key Concepts

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

Linear Equations
Linear equations form the foundation of algebra and represent relationships between variables with straight lines when graphed. For our exercise, x + 5y = 20 is a linear equation since its graph will be a straight line. Every linear equation in two variables can be written in the slope-intercept form, which is expressed as y = mx + b.

In this form, m symbolizes the slope, which shows the line's steepness, and b represents the y-intercept, which is where the line crosses the y-axis. Converting equations into this form not only allows us to quickly identify these key characteristics of the line but also simplifies graphing and comparing different linear equations.
Slope of a Line
The slope is a measure of how steep a line is. In the context of the problem, once the equation is in slope-intercept form, we can easily spot the slope as the coefficient of x. For the equation y = -1/5 * x + 4, the slope is -1/5. This number tells us that for every five units you move to the right along the x-axis, the line will drop by one unit. If the slope is negative, as it is in our case, the line falls as it moves from left to right. If it were positive, the line would rise. A zero slope means the line is horizontal, and an undefined slope (results from division by zero) indicates a vertical line.
Y-intercept
The y-intercept is the point where the line crosses the y-axis. To find it, look at the constant term of the equation once it is in slope-intercept form. For the provided equation, y = -1/5 * x + 4, the y-intercept is 4. This means that our line will cross the y-axis at the point (0, 4). When a line's equation is correctly rearranged, this value is always the lone number without an x next to it in the slope-intercept equation. Identifying the y-intercept is crucial for graphing since it provides a starting point and, combined with the slope, it helps us to draw the entire line.

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Most popular questions from this chapter

Show that the distance between the point \(\left(x_{1}, y_{1}\right)\) and the line \(A x+B y+C=0\) is Distance \(=\frac{\left|A x_{1}+B y_{1}+C\right|}{\sqrt{A^{2}+B^{2}}}\)

Determine whether the function is even, odd, or neither. Use a graphing utility to verify your result. $$f(x)=\sin ^{2} x$$

Find the domain of the function. $$f(x)=\sqrt{x}+\sqrt{1-x}$$

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(f(x)=f(-x)\) for all \(x\) in the domain of \(f\), then the graph of \(f\) is symmetric with respect to the \(y\) -axis.

Prove that the product of two even (or two odd) functions is even.
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