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Chapter 8: Problem 61

Write an equation in slope-intercept form for the line that satisfies the given conditions. (Lesson \(3-4\) ) \(x\) -intercept is \(2, y\) -intercept \(=-8\)

Short Answer

Expert verified
The equation in slope-intercept form is \( y = 4x - 8 \).

Step by step solution

01

Recall the Slope-Intercept Form

The slope-intercept form of a linear equation is given by \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept of the line.
02

Identify the Y-Intercept

From the problem, the y-intercept \( b \) is given as \(-8\). Thus, our equation starts as \( y = mx - 8 \).
03

Use the X-Intercept to Find the Slope

The x-intercept is given as 2, which means the line passes through the point \((2,0)\). Using this point and the equation so far \( y = mx - 8 \), we substitute \( x = 2 \) and \( y = 0 \): \[ 0 = m(2) - 8 \].
04

Solve for the Slope

Using the equation from Step 3, solve for \( m \):\[ 0 = 2m - 8 \]. Adding 8 on both sides gives \( 8 = 2m \). Divide both sides by 2 to find \( m = 4 \).
05

Write the Final Equation

Substitute \( m = 4 \) and \( b = -8 \) into the slope-intercept form: \( y = 4x - 8 \). This is the equation of the line in slope-intercept form.

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Key Concepts

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

Understanding Linear Equations
Linear equations are simple yet foundational. They describe a straight line on a graph. The basic form of a linear equation is the slope-intercept form: \( y = mx + b \). Here, \( x \) and \( y \) are the variables, \( m \) is the slope, and \( b \) is the y-intercept. What makes linear equations interesting is their application in finding relationships between two variables that change at a constant rate.
Consider them like a recipe for creating a straight line. By plugging in different \( x \) values, you can find corresponding \( y \) values, hence drawing a line. They are fundamental in algebra and help us understand more complex mathematical concepts.
The Role of Slope in Linear Equations
The slope \( m \) in the linear equation \( y = mx + b \) tells us how steep the line is. It shows the rate of change between the variables \( x \) and \( y \). Imagine climbing a hill: the slope represents how steep or gentle the hill is.
  • Positive slope: the line rises as you move from left to right.
  • Negative slope: the line falls as you move from left to right.
  • Zero slope: the line is perfectly horizontal.
  • Undefined slope: the line is perfectly vertical.
If you have a point on the graph, like \((2,0)\) in our exercise, you can use it to find the slope if you know another point or the intercepts. For example, using the x-intercept \((2,0)\) and \(-8\) as the y-intercept, you can calculate the slope, as shown in the solution steps.
Exploring the X-Intercept
The x-intercept is where the line crosses the x-axis. This point is vital because it provides a specific value for \( x \) when \( y = 0 \). Here, the x-intercept is 2. It means the line passes through point \((2,0)\).
To find the equation of a line, knowing the x-intercept helps in determining the slope, especially if you also know the y-intercept. Substituting the x-intercept into the equation lets you solve for the slope, just as demonstrated in the exercise solution.
Remember, the x-intercept can tell us a lot about the behavior of the line in the horizontal direction.
Understanding the Y-Intercept
The y-intercept is the point where the line crosses the y-axis. In the slope-intercept equation \( y = mx + b \), the y-intercept is represented by \( b \). For our problem, \( b = -8 \) indicates that when \( x = 0 \), \( y \) will be \(-8\).
Having the y-intercept allows you to start building the equation of the line. It acts as a starting point, and with the slope, you can determine what other points lie on the line.
It's crucial when plotting a line graphically. Simply find \( b \) on the y-axis and use the slope to find adjacent points. This method will enable you to sketch the entire line with ease.

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