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Chapter 1: Problem 24

Find the \(x\) - and \(y\) -intercepts of the graph of the equation. \(y=8-3 x\)

Short Answer

Expert verified
The x-intercept of the graph of the equation is \(8/3\) and the y-intercept is \(8\).

Step by step solution

01

Find the x-intercept

Set \(y = 0\), and solve for \(x\). In our equation, \(y=8-3x\), set \(y = 0\), which gives us \(0=8-3x\). Solving for \(x\), we add \(3x\) to both sides to leave \(3x\) alone on the right side of the equation to get \(3x = 8\). Then, we divide both sides by 3 to solve for \(x\), which gives us \(x = 8/3\).\nThus, the x-intercept is \(8/3\).
02

Find the y-intercept

Now we will find the y-intercept by setting \(x = 0\). In the equation, \(y=8-3x\), set \(x = 0\), which results in \(y=8-3(0) = 8\). Therefore, the y-intercept is \(8\).

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Key Concepts

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

Understanding the x-intercept
Finding the x-intercept of a linear equation involves determining where the graph of the equation crosses the x-axis. This happens when the y-coordinate is zero. In simpler terms, it's the point where the graph touches the horizontal axis.
To find the x-intercept, we set the equation equal to zero
  • For example, with the equation \(y = 8 - 3x\), we replace \(y\) with zero to find where the plot intersects the x-axis.
  • This gives us the equation \(0 = 8 - 3x\).
  • We solve for \(x\) by isolating the variable on one side. First, add \(3x\) to both sides resulting in \(3x = 8\), and then divide by 3 to get \(x = \frac{8}{3}\).
So, the x-intercept of the equation \(y = 8 - 3x\) is \(x = \frac{8}{3}\). This means that if you were to graph it, the line would cross the x-axis at \(x = \frac{8}{3}\). The capabilities of calculating x-intercepts is particularly useful in various real-world applications such as predicting profits or when plotting trajectories.
Unveiling the y-intercept
The y-intercept is a key component of graphing linear equations and refers to the point where the line intersects the y-axis. This occurs when the x-coordinate is zero, representing the position on the vertical axis.
Determining the y-intercept is straightforward:
  • We start with our equation, \(y = 8 - 3x\).
  • Then, set \(x = 0\) and calculate the value of \(y\).
  • Substituting, we have \(y = 8 - 3(0)\).
  • The simplified result is \(y = 8\).
Therefore, the y-intercept in this linear equation is 8. On a graph, this indicates the line crosses the y-axis at \(y = 8\). Knowing the y-intercept helps in forecasting scenarios where initial values are needed, and it also aids in understanding the line's vertical positioning.
Solving linear equations made easy
Linear equations form the backbone of algebra and describe a straight line. To solve a linear equation, your goal is to find the values of variables that make the equation true. These equations can illustrate numerous relationships, including rates and surfaces.
Here's a simple guide to getting it done:
  • A linear equation such as \(y = 8 - 3x\) includes variables and constants.
  • Start by simplifying both sides of the equation if necessary.
  • Isolate your unknown variable. In step-by-step scenarios, you'll add or subtract terms, and divide or multiply results to untangle \(x\) or \(y\).
  • Check your solution by substituting back into the original equation.
In our previous x-intercept calculation, we practiced solving for \(x\) by reformatting \(0 = 8 - 3x\) and found \(x = \frac{8}{3}\). Solving linear equations is fundamental in higher-level math and practical scenarios such as budgeting or forecasting.

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Most popular questions from this chapter

True or False? In Exercises \(71-74\) , determine whether the statement is true or false. Justify your answer. The graphs of \(f(x)=|x|+6\) and \(f(x)=|-x|+6\) are identical.

Hourly Wage Your wage is \(\$ 10.00\) per hour plus \(\$ 0.75\) for each unit produced per hour. So, your hourly wage \(y\) in terms of the number of units produced \(x\) is \(y=10+0.75 x\) (a) Find the inverse function. What does each variable represent in the inverse function? (b) Determine the number of units produced when your hourly wage is \(\$ 24.25 .\)

Parallel and Perpendicular Lines, determine whether the lines \(L_{1}\) and \(L_{2}\) passing through the pairs of points are parallel, perpendicular, or neither. $$\begin{array}{l}{L_{1} :(3,6),(-6,0)} \\ {L_{2} :(0,-1),\left(5, \frac{7}{3}\right)}\end{array}$$

True or False? In Exercises 95 and \(96,\) determine whether the statement is true or false. Justify your answer. If the inverse function of \(f\) exists and the graph of \(f\) has a \(y\) -intercept, then the \(y\) -intercept of \(f\) is an \(x\) -intercept of \(f^{-1}\) .

A function with a square root cannot have a domain that is the set of real numbers.
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