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Chapter 2: Problem 52

Use a graphing calculator to graph each equation, and then find the \(x\) -coordinate of the \(x\) -intercept to the nearest hundredth. Determining the \(x\) -intercepts of a Graph. \(0.3 x+y=7.5\)

Short Answer

Expert verified
The x-intercept is at x = 25.

Step by step solution

01

Rewrite the Equation in Slope-Intercept Form

Start by rewriting the equation in the form of \(y = mx + b\), which is the slope-intercept form. The given equation is \(0.3x + y = 7.5\). Subtract \(0.3x\) from both sides to isolate \(y\), getting \(y = -0.3x + 7.5\).
02

Understand the Concept of x-Intercept

The \(x\)-intercept is the point where the graph of the equation crosses the \(x\)-axis. At this point, the value of \(y\) is zero. Therefore, to find the \(x\)-intercept, set \(y = 0\) in the equation and solve for \(x\).
03

Solve for the x-Intercept

Using the equation \(y = -0.3x + 7.5\), set \(y = 0\) to find the \(x\)-intercept: \(0 = -0.3x + 7.5\). Solving for \(x\), we add \(0.3x\) to both sides: \(0.3x = 7.5\).Divide both sides by \(0.3\): \(x = \frac{7.5}{0.3} = 25\).
04

Graph the Equation using a Graphing Calculator

Enter the equation \(y = -0.3x + 7.5\) into the graphing calculator. Look at the graph to confirm that it intersects the \(x\)-axis at \(x = 25\). This visual confirmation ensures that our calculations are correct.
05

Verify the x-Intercept Value

To ensure accuracy, re-evaluate the calculations and check the graph's intersection with the \(x\)-axis. Our calculation shows the correct x-intercept at \(x = 25\).

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Key Concepts

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

Using a Graphing Calculator Effectively
A graphing calculator is a powerful tool that can help visualize the relationship in a linear equation. It is especially useful when you want to find the x-intercept of a graph. To use it effectively, follow these steps:
  • Enter the equation in the format the calculator accepts, often in slope-intercept form.
  • Make sure the window settings are appropriate so that the x-axis and y-axis are clearly visible.
  • Graph the equation and observe where the line crosses the x-axis. This point is your x-intercept.
Once you've plotted the equation, you can visually confirm the calculated intercept. This visual reference can reinforce understanding and ensure that calculations done by hand or with algebra verify the observed point on the graph.
Understanding Slope-Intercept Form
The slope-intercept form of a linear equation is one of the most straightforward forms to use, expressed as \( y = mx + b \). Here, \( m \) represents the slope, and \( b \) is the y-intercept. Let's break this down further:
  • Slope \( m \): This indicates the steepness of the line and the direction it goes. A positive slope means the line rises as it moves from left to right, while a negative slope means it falls.
  • Y-intercept \( b \): The y-intercept is where the line crosses the y-axis. This tells us the starting point of the line when \( x = 0 \).
Using the slope-intercept form makes graphing easier and helps quickly determine the key characteristics of a line, such as intercepts and slope without additional calculations. This form is highly efficient when solving equations involving lines.
Solving Linear Equations for the x-Intercept
Knowing how to find the x-intercept of a linear equation is crucial, especially in various applications like physics or economics. Here's how you solve for the x-intercept, step-by-step:
  • Start with the equation in slope-intercept form, \( y = mx + b \).
  • Set \( y = 0 \) because the x-intercept is where the line crosses the x-axis, and at this point, \( y \) is always zero.
  • Solve for \( x \):
    • Use algebraic steps to isolate \( x \), first by eliminating any constants or coefficients from one side.
    • Simplify the equation to solve for \( x \), providing the exact location of the x-intercept on the graph.
Solving the equation typically involves basic algebraic manipulation, ensuring you understand each transformation step thoroughly. This approach not only finds the x-intercept but reinforces how various algebraic principles come together.

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Most popular questions from this chapter

The function $$T(a)=837.50+0.15(a-8,375)$$ (where \(a\) is adjusted gross income) is a model of the instructions given on the first line of the following tax rate Schedule X. a. Find \(T(25,000)\) and interpret the result. b. Write a function that models the second line on Schedule X.

Windchill. A combination of cold and wind makes a person feel colder than the actual temperature. The table shows what temperatures of \(35^{\circ} \mathrm{F}\) and \(15^{\circ} \mathrm{F}\) feel like when a 15 -mph wind is blowing. The relationship between the actual temperature and the windchill temperature can be modeled with a linear equation. a. Write the equation that models this relationship. Answer in slope-intercept form. b. What information is given by the \(y\) -intercept of the graph of the equation found in part (a)? $$ \begin{array}{|c|c|} \hline \multicolumn{2}{|c|}\text { 15-mph wind } \\ \hline \text { Actual temperature } \boldsymbol{x} & \text { Windchill temperature } \boldsymbol{y} \\ \hline 35^{\circ} \mathrm{F} & 25^{\circ} \mathrm{F} \\ \hline 15^{\circ} \mathrm{F} & 0^{\circ} \mathrm{F} \\ \hline \end{array} $$

Write an equation for a linear function whose graph has the given characteristics. Passes through \((-2,2)\) and \((2,-8)\)

Let \(f(x)=-2 x+5 .\) For what value of \(x\) does function \(f\) have the given value? \(f(x)=-7\)

Find \(h(5)\) and \(h(-2)\). \(h(x)=\frac{x^{2}+x-2}{x^{2}-5 x}\)
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