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Chapter 3: Problem 26

(a) complete the given table for each equation and then (b) graph the equation. $$ \begin{aligned} &x-y=5\\\ &\begin{array}{c|c} {x} & {y} \\ \hline 0 & {} \\ \hline & {0} \\ \hline 1 & {} \\ \hline 3 & {} \end{array} \end{aligned} $$

Short Answer

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Step by step solution

01

Understand the Equation

The given equation is \( x - y = 5 \). Rewrite the equation to express \( y \) in terms of \( x \): \( y = x - 5 \).
02

Fill in the Table for Given x-values

Substitute the given \( x \)-values into the equation \( y = x - 5 \) to find the corresponding \( y \)-values:
03

Substitute \( x = 0 \)

For \( x = 0 \): \

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Key Concepts

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

linear equations
Linear equations are fundamental in algebra. They form a straight line when graphed on a coordinate plane. A linear equation typically looks like ax + by = c where a , b , and c are constants. These equations model relationships with constant rates of change. They are useful for various real-life applications, such as calculating distances, speeds, and financial predictions. Understanding linear equations is crucial because they form the basis for more complex topics in mathematics.
solving for y
To graph a linear equation, we often need to solve for y . This makes it easier to identify and plot points on a graph. For example, given the equation x - y = 5 , solving for y means we need to isolate y on one side of the equation. We can do this by rearranging the terms: y = x - 5 . This form is known as the slope-intercept form because it clearly displays the slope (the number in front of x ) and the y-intercept (the constant term). Getting the equation into this form is a critical step in graphing.
graphing equations
Graphing equations visually demonstrates the relationship between x and y . For linear equations, their graphs will always be straight lines. Here's how you can graph the equation y = x - 5 :
  • First, create a table of values by choosing several x -values.
  • Substitute these values into the equation to find the corresponding y -values.
  • Plot these (x, y) points on a coordinate grid.
  • Draw a line through these points.
For instance, if x = 0 , then y = -5 . Plot the point (0, -5) . If x = 1 , then y = -4 , and so on. Connecting these points will give you a straight line.
table of values
Creating a table of values helps in organizing the solution process. It enables us to see how different x -values affect y . For y = x - 5 , substituting values:
  • If x = 0 , y = 0 - 5 = -5
  • If x = 1 , y = 1 - 5 = -4
  • If x = 3 , y = 3 - 5 = -2
These ordered pairs (x, y) give exact points to plot on the coordinate grid, smoothing the graphing process. By filling out the table methodically, graphing becomes straightforward and accurate.

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Most popular questions from this chapter

For each situation, (a) write an equation in the form \(y=m x+b,(b)\) find and interpret the ordered pair associated with the equation for \(x=5,\) and \((c)\) answer the question. A rental car costs \(\$ 50\) plus \(\$ 0.20\) per mile. Let \(x\) represent the number of miles driven and \(y\) represent the total charge to the renter. How many miles was the car driven if the renter paid \(\$ 84.60 ?\)

A wholesaler of premium organic planting mix notices that the retail garden centers are not buying her product because of its high price of \(\$ 1.57\) per cubic foot. She decides to mix sawdust with the planting mix to lower the price per cubic foot. If the wholesaler can buy the sawdust for \(\$ 0.10\) per cubic foot, how many cubic feet of each must be mixed to have \(6,000\) cubic feet of planting mix that could be sold to retailers for \(\$ 1.08\) per cubic foot?

Find an equation of the line that satisfies the given conditions. (a) Write the equation in slope-intercept form. (b) Write the equation in standard form. Through \((-1,3) ;\) parallel to \(-x+3 y=12\)

An equation that defines \(y\) as a function fof \(x\) is given. (a) Solve for \(y\) in terms of \(x,\) and \(r e-\) place \(y\) with the function notation \(f(x) .\) (b) Find \(f(3) .\) See Example 6. $$ x-4 y=8 $$

Solve each problem. Forensic scientists use the lengths of certain bones to calculate the height of a person. Two bones often used are the tibia \((t),\) the bone from the ankle to the knee, and the femur \((r),\) the bone from the knee to the hip socket. A person's height ( \(h\) ) in centimeters is determined from the lengths of these bones by using functions defined by the following formulas. For men: \(h(r)=69.09+2.24 r\) or \(h(t)=81.69+2.39 t\) For women: \(h(r)=61.41+2.32 r\) or \(h(t)=72.57+2.53 t\) A.Find the height of a man with a femur measuring \(56 \mathrm{cm} .\) B. Find the height of a man with a tibia measuring \(40 \mathrm{cm} .\) C. Find the height of a woman with a femur measuring \(50 \mathrm{cm} .\) D. Find the height of a woman with a tibia measuring \(36 \mathrm{cm} .\) (PICTURE CANT COPY)
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