Chapter 3

Solve each system by elimination. \(\left\\{\begin{array}{l}{2 x-3 y=-1} \\ {3 x+4 y=8}\end{array}\right.\)

Graph and solve each system. Where necessary, estimate the solution. $$ \left\\{\begin{array}{l}{-x-2=-2 y} \\ {2 x-4 y-4=0}\end{array}\right. $$

Solve each system. $$ \left\\{\begin{array}{l}{x+6 z=12} \\ {-2 x+3 y=6} \\\ {y-\frac{z}{2}=\frac{5}{2}}\end{array}\right. $$

Evaluate each expression for \(a=3\) and \(b=-5\). \(-4+2 a b\)

Geography Use the map to identify the geographic feature found at each location. latitude \(3^{\circ} \mathrm{S}\) longitude \(37^{\circ} \mathrm{E}\) elevation \(19,340 \mathrm{ft}\)

Solve each system by elimination. \(\left\\{\begin{array}{l}{5 x-2 y=-19} \\ {2 x+3 y=0}\end{array}\right.\)

Banking To pay your monthly bills, you can either open a checking account or use an online banking service. A local bank charges \(\$ 3\) per month and \(\$ .40\) per check, while an online services charges a flat fee of \(\$ 9\) per month. a. Write and graph a system of linear equations to model the cost \(c\) of each service for \(b\) bills that you need to pay monthly. b. Find the point of intersection of the two linear models. What does this answer represent? c. If you pay about 12 bills per month, which service should you choose? Explain.

Solve each system. $$ \left\\{\begin{array}{l}{4 x-y+z=-5} \\ {-x+y-z=5} \\ {2 x-z-1=y}\end{array}\right. $$

Evaluate each expression for \(a=3\) and \(b=-5\). \(a+\frac{3 b}{a}\)

Solve each system by elimination. \(\left\\{\begin{aligned} r+3 s &=7 \\ 2 r-s &=7 \end{aligned}\right.\)

Classify each system without graphing. $$ \left\\{\begin{array}{l}{3 x-2 y=8} \\ {4 y=6 x-5}\end{array}\right. $$

History Exercises 39 and 40 appeared in the book Algebrical Problems, published in \(1824 .\) Write and solve a system for each problem. Ten apples cost a penny, and 25 pears cost two pennies. Suppose I buy 100 apples and pears for 9\(\frac{1}{2}\) pennies. How many of each shall I have?

Evaluate each expression for \(a=3\) and \(b=-5\). 3\((a-b)\)

Solve each system by elimination. \(\left\\{\begin{array}{l}{y=4-x} \\ {3 x+y=6}\end{array}\right.\)

Classify each system without graphing. $$ \left\\{\begin{array}{l}{2 x+8 y=6} \\ {x=-4 y+3}\end{array}\right. $$

History Exercises 39 and 40 appeared in the book Algebrical Problems, published in \(1824 .\) Write and solve a system for each problem. A fish was caught whose tail weighed 9 lb. Its head weighed as much as its tail plus half its body. Its body weighed as much as its head and tail. What did the fish weigh?

Evaluate each expression for \(a=3\) and \(b=-5\). \(4 a-2+3 b\)

Classify each system without graphing. $$ \left\\{\begin{array}{l}{3 a+6 b=14} \\ {-a+2 b=3}\end{array}\right. $$

Open-Ended Write and graph a system of inequalities for which the solution is bounded by a dashed vertical line and a solid horizontal line.

Evaluate each expression for \(a=3\) and \(b=-5\). \(\frac{a-b}{2 a}\)

Solve each system by elimination. \(\left\\{\begin{array}{l}{3 m+4 n=-13} \\ {5 m+6 n=-19}\end{array}\right.\)

Classify each system without graphing. $$ \left\\{\begin{array}{l}{3 m=-5 n+4} \\ {n-\frac{6}{5}=-\frac{3}{5} m}\end{array}\right. $$

Error Analysis A student says that the system consisting of \(x=0, y=0,\) and \(z=0\) has no solutions. Explain the student's error.

Evaluate each expression for \(a=3\) and \(b=-5\). \(b(2 b-a)\)

In a mayoral election, the number of votes for the incumbent was 25\(\%\) more than the number for the opponent. Altogether, the two candidates received 5175 votes. How many votes did the incumbent mayor receive?

Classify each system without graphing. $$ \left\\{\begin{array}{l}{-12 x+4 y=8} \\ {y-4=3 x}\end{array}\right. $$

Solve each system of inequalities by graphing. $$ \left\\{\begin{array}{l}{x+y<8} \\ {x \geq 0} \\ {y \geq 0}\end{array}\right. $$

Classify each system without graphing. $$ \left\\{\begin{array}{l}{-6 y+18=12 x} \\ {3 y+6 x=9}\end{array}\right. $$

Open-Ended Write your own system having three variables. Begin by choosing the solution. Then write three equations that are true for your solution. Use elimination to solve the system.

Solve each system of inequalities by graphing. $$ \left\\{\begin{array}{l}{2 y-4 x \leq 0} \\ {x \geq 0} \\ {y \geq 0}\end{array}\right. $$

Solve each system. \(\left\\{\begin{array}{l}{5 x+y=0} \\ {5 x+2 y=30}\end{array}\right.\)

Fees Suppose you are going on vacation and leaving your dog in a kennel.The Bowowery charges \(\$ 5\) per day, which includes a one-time grooming treatment. The Poochpad charges \(\$ 20\) per day and a one-time fee of \(\$ 30\) for grooming. a. Write a system of equations to represent the cost \(c\) for \(d\) days that your dog will stay at a kennel. b. Using a graphing calculator, find the number of days for which the costs are the same. c. If your vacation is a week long, which kennel should you choose? Explain.

Geometry In the regular polyhedron described below, all faces are congruent polygons. Use a system of three linear equations to find the numbers of vertices, edges, and faces. Every face has five edges and every edge is shared by two faces. Every face has five vertices and every vertex is shared by three faces. The sum of the number of vertices and faces is two more than the number of edges.

Solve each system of inequalities by graphing. $$ \left\\{\begin{array}{l}{y \geq-2 x+4} \\ {x>-3} \\ {y \geq 1}\end{array}\right. $$

Solve each system. \(\left\\{\begin{array}{l}{2 m=-4 n-4} \\ {3 m+5 n=-3}\end{array}\right.\)

What is the solution of the system? \(\left\\{\begin{array}{cc}{-3 x+2 y-z=} & {6} \\ {3 x+y+2 z=} & {5} \\ {2 x-2 y-z=} & {-5}\end{array}\right.\) $$ \begin{array}{ll}{\text { A. }(6,5,-3)} & {\text { B. }(1,4,-1)} \\ {\text { C. }\left(0, \frac{17}{5}, \frac{4}{5}\right)} & {\text { D. no solution }}\end{array} $$

Solve each system of inequalities by graphing. $$ \left\\{\begin{array}{l}{y \leq \frac{2}{3} x+2} \\ {y \geq|x|+2}\end{array}\right. $$

Solve each system. \(\left\\{\begin{array}{l}{7 x+2 y=-8} \\ {8 y=4 x}\end{array}\right.\)

What is the solution of the system? \(\left\\{\begin{array}{l}{x+3 y-2 z=-8} \\\ {3 x-y+z=11} \\ {2 x+4 y+2 z=14}\end{array}\right.\) $$ \begin{array}{ll}{\text { F. }(2,0,5)} & {\text { G. }(-8,11,14)} \\ {\text { H. }\left(-2, \frac{4}{3}, 5\right)} & {\text { J. no solution }}\end{array} $$

Solve each system of inequalities by graphing. $$ \left\\{\begin{array}{l}{y < x-1} \\ {y > -|x-2|+1}\end{array}\right. $$

Error Analysis A student claims to find the \(x\) -intercept of a plane by substituting 0 for \(x\) in the equation of the plane. Explain the student's error.

Solve each system. \(\left\\{\begin{array}{l}{v=9 t+300} \\ {v=7 t+400}\end{array}\right.\)

What is the solution of the system? \(\left\\{\begin{array}{l}{y=-2 x+10} \\\ {-x+y-2 z=-2} \\ {3 x-2 y+4 z=7}\end{array}\right.\) $$ \begin{array}{ll}{\text { A. }\left(3,-4, \frac{3}{2}\right)} & {\text { B. }\left(3,16, \frac{15}{2}\right)} \\ {\text { C. }\left(-3,16, \frac{15}{2}\right)} & {\text { D. }\left(3,4, \frac{3}{2}\right)}\end{array} $$

Solve each system of inequalities by graphing. $$ \left\\{\begin{array}{l}{2 x+y \leq 3} \\ {y>|x+3|-2}\end{array}\right. $$

a. Geometry Use the Pythagorean Theorem to find the distance between \((1,2,4)\) and \((3,-2,7)\) . (Hintt Recall the Distance Formula.) b. Make a Conjecture Make a conjecture about how to find the coordinates of the midpoint of a segment in coordinate space.

Solve each system. \(\left\\{\begin{aligned} 80 x+60 y &=85 \\ 100 x-40 y &=20 \end{aligned}\right.\)

Open-Ended Write a second equation for each system so that the system will have the indicated number of solutions. $$ \begin{array}{l}{\text { an infinite number }} \\ {\left\\{\begin{array}{c}{3 y=6 x+7} \\ {?}\end{array}\right.}\end{array} $$

Solve each system. \(\left\\{\begin{array}{l}{2 x+3 y=0} \\ {7 x=3(2 y)+3}\end{array}\right.\)

Reasoning Is it possible for an inconsistent linear system to consist of two lines with the same \(y\) -intercept? Explain.

Graph each equation in three-dimensional coordinate space. \(x=3\)