Chapter 3

Solve and graph the solution set on a number line: \(2 x-3 \leq 5 .\) (Section \(2.7,\) Example 6 )

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 \(x-2 y=6 ?\)

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

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Every line in the rectangular coordinate system has an equation that can be expressed in slope- intercept form.

Exercises \(75-77\) will help you prepare for the material covered in the next section. From \((0,-3),\) move 4 units up and 1 unit to the right. What point do you obtain?

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 \(x-2 y=6 ?\)

A new car worth \(\$ 24,000\) is depreciating in value by \(\$ 3000\) per year. The mathematical model $$y=-3000 x+24,000$$ 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.

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

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 \(3 x+2 y=5\) has slope \(-\frac{3}{2}\).

Exercises \(75-77\) will help you prepare for the material covered in the next section. From \((0,1),\) move 2 units down and 3 units to the right. What point do you obtain?

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.

A new car worth \(\$ 45,000\) is depreciating in value by \(\$ 5000\) 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.

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

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 \(2 y=3 x+7\) has a \(y\) -intercept of 7.

Exercises \(75-77\) will help you prepare for the material covered in the next section. Solve for \(y: 2 x+5 y=0\)

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

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

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. 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.

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

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

\( \quad \frac{x}{2}+7=13-\frac{x}{4}\)

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

write each sentence as a linear equation in two variables. Then graph the equation. $$y=3, \text { or } y=0 x+3$$

$$\text { Simplify: } \quad 3\left(12 \div 2^{2}-3\right)^{2}$$.

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

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

14 is \(25 \%\) of what number?

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

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

Solve for \(y\) and put the equation in slope-intercept form. $$y-3=4(x+1)$$

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

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

Solve for \(y\) and put the equation in slope-intercept form. $$y+3=-\frac{3}{2}(x-4)$$

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

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.

Solve for \(y\) and put the equation in slope-intercept form. $$y-30.0=0.265(x-10)$$

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

At the beginning of a semester, a student purchased cight pens and six pads for a total cost of \(\$ 14.50 .\) 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 \(\$ 0.75\) each, find the cost of one pen.

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

A nursery offers a package of three small orange trees and four small grapefruit trees for \(\$ 22\). 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 \(\$ 2.50,\) 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.

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.

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.

From 1997 through \(2007,\) the federal minimum wage remained constant at \(\$ 5.15\) per hour, so I modeled the situation with \(y=5.15\) and the graph of a vertical line.

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 \(=4\)

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 cducation 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 vicwed 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 \(S=100\)

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. \(2 x+y=4\)

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 qualificd 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. \(3 x-y=9\)