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Chapter 4: Problem 299

Identify the slope and \(y\) -intercept of each line. \(4 x+y=8\)

Short Answer

Expert verified
The slope is -4 and the y-intercept is 8.

Step by step solution

01

Rewrite the equation in slope-intercept form

The slope-intercept form of a line is given by \(y = mx + b\), where m represents the slope and b represents the y-intercept. Start by rewriting the given equation, \(4x + y = 8\), in this form.
02

Isolate y on one side

To isolate y, subtract 4x from both sides of the equation: \[4x + y - 4x = 8 - 4x\]which simplifies to\[y = -4x + 8\]
03

Identify the slope and y-intercept

Now that the equation is in the form \(y = mx + b\), compare it to \(y = -4x + 8\). The slope (m) is -4, and the y-intercept (b) is 8.

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Key Concepts

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

slope calculation
When working with linear equations, calculating the slope is essential. The slope, represented by the letter \(m\), indicates the steepness or incline of the line. It can be thought of as the 'rise over run', or how much the line goes up or down as you move along it.

Let's say you have an equation in slope-intercept form, \(y = mx + b\). To find the slope:
  • Rewrite the equation in this form (if it鈥檚 not already).
  • The coefficient of \(x\) (the number directly before \(x\)) is the slope.
For example, in the equation \(y = -4x + 8\), the slope \(m\) is -4. This means for every unit you move to the right along the x-axis, the line goes down by 4 units. Easy, right? Just remember: the slope tells you how angled the line is!
y-intercept identification
Identifying the y-intercept is another crucial aspect of understanding linear equations. The y-intercept, represented by the letter \(b\), is the point where the line crosses the y-axis. This is where the value of \(x\) is zero. In simpler terms, it tells you what \(y\) equals when \(x\) is zero.

To find the y-intercept in the slope-intercept form \(y = mx + b\):
  • First, ensure your equation is written in the slope-intercept form.
  • Look at the constant term to identify the y-intercept.
For example, in \(y = -4x + 8\), the y-intercept \(b\) is 8. This means that when the line crosses the y-axis, the value of \(y\) is 8. By knowing the y-intercept, you can quickly plot this point on the graph and start drawing your line.
linear equations
Linear equations are fundamental in algebra and represent straight lines when plotted on a graph. The most common form is the slope-intercept form given by \(y = mx + b\). Here鈥檚 a quick rundown:
  • \(y\) stands for the dependent variable
  • \(x\) represents the independent variable
  • \(m\) is the slope, showing how steep the line is
  • \(b\) is the y-intercept, where the line crosses the y-axis
Linear equations describe relationships where one variable changes at a constant rate with respect to the other. For example, in the equation \(4x + y = 8\), rearranging it to the form \(y = -4x + 8\) shows us that:
  • The slope \(m = -4\), indicating that for each unit increase in \(x\), \(y\) decreases by 4 units.
  • The y-intercept \(b = 8\), showing the line crosses the y-axis at \(8\).
These key characteristics make linear equations very straightforward to understand and work with.

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Most popular questions from this chapter

Find an equation of a line perpendicular to the given line and contains the given point. Write the equation in slope-intercept form. line \(y-6=0,\) point 4(-5,-3)

Find the equation of a line containing the given points. Write the equation in slope-intercept form. (3,-4) and (5,-4)

Graph the linear inequality \(y>x\)

Graph the linear inequality \(y \leq-x\)

Find an equation of a line perpendicular to the given line and contains the given point. Write the equation in slope-intercept form. . line \(4 x+5 y=-3\), point (0,0)
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