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Chapter 4: Problem 301

Identify the slope and \(y\) -intercept of each line. \(8 x+3 y=12\)

Short Answer

Expert verified
Slope: \(-\frac{8}{3}\), y-intercept: \(4\)

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 \(m\) is the slope and \(b\) is the y-intercept. Start by isolating \(y\) in the given equation \(8x + 3y = 12\).
02

Isolate the term containing y

Subtract \(8x\) from both sides to get: \[3y = -8x + 12\]
03

Solve for y

Divide both sides of the equation by 3 to solve for \(y\): \[y = -\frac{8}{3}x + \frac{12}{3}\]
04

Simplify the equation

Simplify \(\frac{12}{3}\) to get: \[y = -\frac{8}{3}x + 4\]
05

Identify the slope and y-intercept

Compare the equation \(y = -\frac{8}{3}x + 4\) to the slope-intercept form \(y = mx + b\). The slope \(m\) is \(-\frac{8}{3}\), 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.

slope-intercept form
To understand the slope and y-intercept of a line, we use the slope-intercept form of a linear equation, which is written as \(\text{y = mx + b}\). In this equation:
  • \(m\) represents the slope of the line.
  • \(b\) is the y-intercept, which is the point where the line crosses the y-axis.
Using this form makes it easy to quickly identify both the slope and the y-intercept.

The first step in many linear equations problems is to convert the given equation into this slope-intercept form. This usually involves isolating the variable y on one side of the equation.
linear equations
Linear equations represent straight lines when graphed on a coordinate plane. These equations have no exponents higher than one. For example, the equation \(8x + 3y = 12\) is a linear equation.
To work with linear equations, you often need to rearrange or simplify them to better identify key values such as the slope and y-intercept.
These steps often include isolating the y variable to put the equation into the slope-intercept form.

Rewriting the equation in this way helps us easily compare it to the general form \(y = mx + b\).
isolate variable
Isolating the variable is a crucial step in solving linear equations. To isolate y in an equation like \(8x + 3y = 12\), we perform these steps:
  • First, subtract \(8x\) from both sides to move all x terms to the other side of the equation: \[3y = -8x + 12\]
This step makes it easier to solve for y.
Next, we divide by 3 to get y by itself:
  • \[y = -\frac{8}{3}x + \frac{12}{3}\]
By isolating y, we can clearly see the slope and intercept values in the equation.
simplifying equations
Simplifying equations involves reducing them to their simplest form. This makes it easier to understand and work with.

In the example \(8x + 3y = 12\), we simplify by first subtracting \(8x\) from both sides and then by dividing everything by 3. This gives us:
  • \[y = -\frac{8}{3}x + 4\]
Now, the equation is in slope-intercept form. We can easily see that the slope (\(m\)) is \(-\frac{8}{3}\) and the y-intercept (\(b\)) is 4.
This process often involves:
  • Combining like terms
  • Reducing fractions
  • Rewriting terms to make the equation more comprehensible
It鈥檚 important to practice these steps to become comfortable with simplifying equations.

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Most popular questions from this chapter

Find the equation of each line. Write the equation in slope-intercept form. Parallel to the line \(2 x+3 y=6\), containing point (0,5)

Graph the linear inequality \(y<-3 x-4\)

Money. Gerry wants to have a maximum of $$\$ 100$$ cash at the ticket booth when his church carnival opens. He will have $$\$ 1$$ bills and $$\$ 5$$ bills. If \(x\) is the number of $$\$ 1$$ bills and y is the number of $$\$ 5$$ bills, the inequality $$x+5 y \leq 100$$ models the situation. (a) Graph the inequality. (b) List three solutions to the inequality \(x+5 y \leq 100\) where both x and y are integers.

Find the equation of each line. Write the equation in slope-intercept form. Containing the points (-2,0) and (-3,-2)

Find an equation of a line parallel to the given line and contains the given point. Write the equation in slope-intercept form. line \(3 x-y=4,\) point (3,1)
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