Chapter 3

Graph each linear equation in two variables. Find at least five solutions in your table of values for each equation. $$y=-\frac{5}{2} x+1$$

A new car worth 45,000 dollars is depreciating in value by 5000 dollars per year. The mathematical model $$y=-5000 x+45,000$$ describes the car's value, \(y,\) in dollars, after \(x\) years. a. Find the \(x\) -intercept. Describe what this means in terms of the car's value. b. Find the \(y\) -intercept. Describe what this means in terms of the car's value. c. Use the intercepts to graph the linear equation. Because \(x\) and \(y\) must be nonnegative (why?), limit your graph to quadrant I and its boundaries. d. Use your graph to estimate the car's value after five years.

Will help you prepare for the material covered in the next section. Solve for \(y: 2 x+5 y=0\)

Graph each linear equation in two variables. Find at least five solutions in your table of values for each equation. $$y=x+\frac{1}{2}$$

What is an \(x\) -intercept of a graph?

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The line whose equation is \(y-3=7(x+2)\) passes through \((-3,2)\)

The relationship between Celsius temperature, \(C,\) and Fahrenheit temperature, \(F\), can be described by a linear equation in the form \(F=m C+b .\) The graph of this equation contains the point \((0,32):\) Water freezes at \(0^{\circ} \mathrm{C}\) or at \(32^{\circ} \mathrm{F}\). The line also contains the point \((100,212):\) Water boils at \(100^{\circ} \mathrm{C}\) or at \(212^{\circ} \mathrm{F}\). Write the linear equation expressing Fahrenheit temperature in terms of Celsius temperature.

Graph each linear equation in two variables. Find at least five solutions in your table of values for each equation. $$y=x-\frac{1}{2}$$

What is a \(y\) -intercept of a graph?

$$\text { Solve: } \frac{x}{2}+7=13-\frac{x}{4} . \text { (Section 2.3, Example 4) }$$

Graph each linear equation in two variables. Find at least five solutions in your table of values for each equation. $$y=4, \text { or } y=0 x+4$$

If you are given an equation of the form \(A x+B y=C\) explain how to find the \(x\) -intercept.

Simplify: \(\quad 3\left(12 \div 2^{2}-3\right)^{2}\). (Section \(1.8,\) Example 6 )

Graph each linear equation in two variables. Find at least five solutions in your table of values for each equation. $$y=3, \text { or } y=0 x+3$$

If you are given an equation of the form \(A x+B y=C\) explain how to find the \(y\) -intercept.

Excited about the success of celebrity stamps, post office officials were rumored to have put forth a plan to institute two new types of thermometers. On these new scales, \(^{\circ} E\) represents degrees Elvis and \(^{\circ} M\) represents degrees Madonna. If it is known that \(40^{\circ} E=25^{\circ} M, 280^{\circ} E=125^{\circ} M\) and degrees Elvis is linearly related to degrees Madonna, write an equation expressing \(E\) in terms of \(M\).

14 is \(25 \%\) of what number? (Section \(2.4,\) Example 6 )

Write each sentence as a linear equation in two variables. Then graph the equation. The \(y\) -variable is 3 more than the \(x\) -variable.

Explain how to graph \(A x+B y=C\) if \(C\) is not equal to zero.

Use a graphing utility to graph \(y=1.75 x-2 .\) Select the best viewing rectangle possible by experimenting with the range settings to show that the line's slope is \(\frac{7}{4}\).

Exercises \(82-84\) will help you prepare for the material covered in the next section. In each exercise, solve for \(y\) and put the equation in slope- intercept form. $$y-3=4(x+1)$$

Write each sentence as a linear equation in two variables. Then graph the equation. The \(y\) -variable exceeds the \(x\) -variable by 4

Explain how to graph a linear equation of the form \(A x+B y=0\).

Exercises \(82-84\) will help you prepare for the material covered in the next section. In each exercise, solve for \(y\) and put the equation in slope- intercept form. $$y+3=-\frac{3}{2}(x-4)$$

Write each sentence as a linear equation in two variables. Then graph the equation. The \(y\) -variable exceeds twice the \(x\) -variable by 5

How many points are needed to graph a line? How many should actually be used? Explain.

How many sheets of paper, weighing 2 grams each, can be put in an envelope weighing 4 grams if the total weight must not exceed 29 grams? (Section \(2.7,\) Example 11 )

Exercises \(82-84\) will help you prepare for the material covered in the next section. In each exercise, solve for \(y\) and put the equation in slope- intercept form. $$y-30.0=0.265(x-10)$$

Write each sentence as a linear equation in two variables. Then graph the equation. The \(y\) -variable is 2 less than 3 times the \(x\) -variable.

Describe the graph of \(y=200\).

List all the natural numbers in this set: $$ \left\\{-2,0, \frac{1}{2}, 1, \sqrt{3}, \sqrt{4}\right\\} $$ (Section \(1.3,\) Example 5 )

At the beginning of a semester, a student purchased eight pens and six pads for a total cost of 14.50 dollars . a. If \(x\) represents the cost of one pen and \(y\) represents the cost of one pad, write an equation in two variables that reflects the given conditions. b. If pads cost 0.75 dollars each, find the cost of one pen.

Describe the graph of \(x=-100\).

Use intercepts to graph \(3 x-5 y=15\) (Section \(3.2,\) Example 4 )

A nursery offers a package of three small orange trees and four small grapefruit trees for 22 dollars. a. If \(x\) represents the cost of one orange tree and \(y\) represents the cost of one grapefruit tree, write an equation in two variables that reflects the given conditions. b. If a grapefruit tree costs 2.50 dollars, find the cost of an orange tree.

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. If I could be absolutely certain that I have not made an algebraic error in obtaining intercepts, I would not need to use checkpoints.

Will help you prepare for the material covered in the first section of the next chapter. Is \((4,-1)\) a solution of both \(x+2 y=2\) and \(x-2 y=6 ?\)

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I like to select a point represented by one of the intercepts as my checkpoint.

Will help you prepare for the material covered in the first section of the next chapter. Is \((-4,3)\) a solution of both \(x+2 y=2\) and \(x-2 y=6 ?\)

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. The graphs of \(2 x-3 y=-18\) and \(-2 x+3 y=18\) must have the same intercepts because I can see that the equations are equivalent.

Will help you prepare for the material covered in the first section of the next chapter. Determine the point of intersection of the graphs of \(2 x+3 y=6\) and \(2 x+y=-2\) by graphing both equations in the same rectangular coordinate system.

Find the coefficients that must be placed in each shaded area so that the equation's graph will be a line with the specified intercepts. \(\square x+\square y=10 ; x\) -intercept \(=5 ; y\) -intercept \(=2\)

Find the coefficients that must be placed in each shaded area so that the equation's graph will be a line with the specified intercepts. \(\square x+\square y=12 ; x\) -intercept \(=-2 ; y\) -intercept \(=4\)

Even as Americans increasingly view a college education as essential for success, many believe that a college education is becoming less available to qualified students. Exercises are based on the data displayed by the graph. (GRAPH CANNOT COPY) The graph shows that in \(2000,31 \%\) of U.S. adults viewed a college education as essential for success. For the period from 2000 through 2010 , the percentage viewing a college education as essential for success increased on average by approximately 2.4 each year. These conditions can be described by the mathematical model $$S=2.4 n+31$$ where \(S\) is the percentage of U.S. adults who viewed college as essential for success \(n\) years after 2000 . a. Let \(n=0,5,10,15,\) and \(20 .\) Make a table of values showing five solutions of the equation. b. Graph the formula in a rectangular coordinate system. Suggestion: Let each tick mark on the horizontal axis, labeled \(n\), represent 5 units. Extend the horizontal axis to include \(n=25 .\) Let each tick mark on the vertical axis, labeled \(S\), represent 10 units and extend the axis to include \(S=100\) c. Use your graph from part (b) to estimate the percentage of U.S. adults who will view college as essential for success in 2018 d. Use the formula to project the percentage of U.S. adults who will view college as essential for success in 2018 .

Use a graphing utility to graph each equation. You will need to solve the equation for \(y\) before entering it. Use the graph displayed on the screen to identify the \(x\) -intercept and the \(y\) -intercept. $$2 x+y=4$$

Even as Americans increasingly view a college education as essential for success, many believe that a college education is becoming less available to qualified students. Exercises are based on the data displayed by the graph. (GRAPH CANNOT COPY) The graph shows that in \(2000,45 \%\) of U.S. adults believed that most qualified students get to attend college. For the period from 2000 through 2010 , the percentage who believed that a college education is available to most qualified students decreased by approximately 1.7 each year. These conditions can be described by the mathematical model $$Q=-1.7 n+45$$ where \(Q\) is the percentage believing that a college education is available to most qualified students \(n\) years after 2000 a. Let \(n=0,5,10,15,\) and \(20 .\) Make a table of values showing five solutions of the equation. b. Graph the formula in a rectangular coordinate system. Suggestion: Let each tick mark on the horizontal axis, labeled \(n\), represent 5 units. Extend the horizontal axis to include \(n=25 .\) Let each tick mark on the vertical axis, labeled \(Q,\) represent 5 units and extend the axis to include \(Q=50\) c. Use your graph from part (b) to estimate the percentage of U.S. adults who will believe that a college education is available to most qualified students in 2018 . d. Use the formula to project the percentage of U.S. adults who will believe that a college education is available to most qualified students in 2018 .

Use a graphing utility to graph each equation. You will need to solve the equation for \(y\) before entering it. Use the graph displayed on the screen to identify the \(x\) -intercept and the \(y\) -intercept. $$3 x-y=9$$

What is the rectangular coordinate system?

Use a graphing utility to graph each equation. You will need to solve the equation for \(y\) before entering it. Use the graph displayed on the screen to identify the \(x\) -intercept and the \(y\) -intercept. $$2 x+3 y=30$$

Explain how to plot a point in the rectangular coordinate system. Give an example with your explanation.