Chapter 10

For Problems 19-48, solve each system by using either the substitution or the elimination-by-addition method, whichever seems more appropriate. (Objective 2) $$ \left(\begin{array}{rr} -(x-6)+6(y+1)= & 58 \\ 3(x+1)-4(y-2) & =-15 \end{array}\right) $$

The income from a student production was $$\$ 47,500$$. The price of a student ticket was $$\$ 15$$, and nonstudent tickets were sold at $$\$ 20$$ each. Three thousand tickets were sold. How many tickets of each kind were sold?

Use Cramer's rule to find the solution set for each of the following systems. (Objective 2) $$ \left(\begin{array}{c} 3 x-\frac{1}{2} y=6 \\ -2 x+\frac{1}{3} y=-4 \end{array}\right) $$

For Problems 19-48, solve each system by using either the substitution or the elimination-by-addition method, whichever seems more appropriate. (Objective 2) $$ \left(\begin{array}{rr} -2(x+2)+4(y-3) & =-34 \\ 3(x+4)-5(y+2) & =23 \end{array}\right) $$

Sue bought 3 packages of cookies and 2 sacks of potato chips for $$\$ 13.50$$. Later she bought 2 more packages of cookies and 5 additional sacks of potato chips for $$\$ 20.00$$. Find the price of a package of cookies.

Explain the difference between a matrix and a determinant.

For Problems 19-48, solve each system by using either the substitution or the elimination-by-addition method, whichever seems more appropriate. (Objective 2) $$ \left(\begin{array}{lr} 5(x+1)-(y+3)= & -6 \\ 2(x-2)+3(y-1)= & 0 \end{array}\right) $$

Give a general description of how to use the substitution method to solve a system of two linear equations in two variables.

Give a step-by-step description of how you would solve the system \(\left(\begin{array}{l}3 x-2 y=7 \\ 5 x+9 y=14\end{array}\right)\) using determinants.

For Problems 19-48, solve each system by using either the substitution or the elimination-by-addition method, whichever seems more appropriate. (Objective 2) $$ \left(\begin{array}{l} 2(x-1)-3(y+2)=30 \\ 3(x+2)+2(y-1)=-4 \end{array}\right) $$

Is it possible for a system of two linear equations in two variables to have exactly two solutions? Defend your answer.

Verify each of the following. The variables represent real numbers. (a) \(\left|\begin{array}{ll}a & b \\ a & b\end{array}\right|=0\) (b) \(\left|\begin{array}{ll}a & a \\ b & b\end{array}\right|=0\) (c) \(\left|\begin{array}{ll}a & b \\ c & d\end{array}\right|=-\left|\begin{array}{ll}b & a \\ d & c\end{array}\right|\) (d) \(\left|\begin{array}{ll}a & b \\ c & d\end{array}\right|=-\left|\begin{array}{ll}c & d \\ a & b\end{array}\right|\) (e) \(k\left|\begin{array}{ll}a & b \\ c & d\end{array}\right|=\left|\begin{array}{ll}k a & b \\ k c & d\end{array}\right|\) (f) \(k\left|\begin{array}{ll}a & b \\ c & d\end{array}\right|=\left|\begin{array}{cc}k a & k b \\ c & d\end{array}\right|\)

For Problems 19-48, solve each system by using either the substitution or the elimination-by-addition method, whichever seems more appropriate. (Objective 2) $$ \left(\begin{array}{l} \frac{1}{2} x-\frac{1}{3} y=12 \\ \frac{3}{4} x+\frac{2}{3} y=4 \end{array}\right) $$

Explain how you would solve the system \(\left(\begin{array}{l}2 x+5 y=5 \\ 5 x-y=9\end{array}\right)\) using the substitution method.

For Problems 19-48, solve each system by using either the substitution or the elimination-by-addition method, whichever seems more appropriate. (Objective 2) $$ \left(\begin{array}{l} \frac{2}{3} x+\frac{1}{5} y=0 \\ \frac{3}{2} x-\frac{3}{10} y=-15 \end{array}\right) $$

For Problems 19-48, solve each system by using either the substitution or the elimination-by-addition method, whichever seems more appropriate. (Objective 2) $$ \left(\begin{array}{l} \frac{2 x}{3}-\frac{y}{2}=-\frac{5}{4} \\ \frac{x}{4}+\frac{5 y}{6}=\frac{17}{16} \end{array}\right) $$

Use your graphing calculator to help determine the solution set for each of the following systems. Be sure to check your answers. (a) \(\left(\begin{array}{l}3 x-y=30 \\ 5 x-y=46\end{array}\right)\) (b) \(\left(\begin{array}{l}1.2 x+3.4 y=25.4 \\ 3.7 x-2.3 y=14.4\end{array}\right)\) (c) \(\left(\begin{array}{l}1.98 x+2.49 y=13.92 \\ 1.19 x+3.45 y=16.18\end{array}\right)\) (d) \(\left(\begin{array}{l}2 x-3 y=10 \\ 3 x+5 y=53\end{array}\right)\) (e) \(\left(\begin{array}{l}4 x-7 y=-49 \\ 6 x+9 y=219\end{array}\right)\) (f) \(\left(\begin{array}{l}3.7 x-2.9 y=-14.3 \\ 1.6 x+4.7 y=-30\end{array}\right)\)

For Problems 19-48, solve each system by using either the substitution or the elimination-by-addition method, whichever seems more appropriate. (Objective 2) $$ \left(\begin{array}{l} \frac{x}{2}+\frac{y}{3}=\frac{5}{72} \\ \frac{x}{4}+\frac{5 y}{2}=-\frac{17}{48} \end{array}\right) $$

For Problems 19-48, solve each system by using either the substitution or the elimination-by-addition method, whichever seems more appropriate. (Objective 2) $$ \left(\begin{array}{l} \frac{3 x+y}{2}+\frac{x-2 y}{5}=8 \\ \frac{x-y}{3}-\frac{x+y}{6}=\frac{10}{3} \end{array}\right) $$

For Problems 19-48, solve each system by using either the substitution or the elimination-by-addition method, whichever seems more appropriate. (Objective 2) $$ \left(\begin{array}{l} \frac{x-y}{4}-\frac{2 x-y}{3}=-\frac{1}{4} \\ \frac{2 x+y}{3}+\frac{x+y}{2}=\frac{17}{6} \end{array}\right) $$

For Problems \(49-60\), solve each problem by setting up and solving an appropriate system of equations. (Objective 3 ) A \(10 \%\)-salt solution is to be mixed with a \(20 \%\)-salt solution to produce 20 gallons of a \(17.5 \%\)-salt solution. How many gallons of the \(10 \%\) solution and how many gallons of the \(20 \%\) solution will be needed?

A small-town library buys a total of 35 books that cost $$\$ 644$$. Some of the books cost $$\$ 16$$ each, and the remainder cost $$\$ 20$$ each. How many books of each price did the library buy?

Suppose that on a particular day the cost of 3 tennis balls and 2 golf balls is $$\$ 7$$. The cost of 6 tennis balls and 3 golf balls is $$\$ 12$$. Find the cost of 1 tennis ball and the cost of 1 golf ball.

For moving purposes, the Hendersons bought 25 cardboard boxes for $$\$ 97.50$$. There were two kinds of boxes; the large ones cost $$\$ 7.50$$ per box, and the small ones were $$\$ 3$$ per box. How many boxes of each kind did they buy?

A motel in a suburb of Chicago rents double rooms for $$\$ 120$$ per day and single rooms for $$\$ 90$$ per day. If a total of 55 rooms were rented for $$\$ 6150$$, how many of each kind were rented?

Suppose that one solution contains \(50 \%\) alcohol and another solution contains \(80 \%\) alcohol. How many liters of each solution should be mixed to make \(10.5\) liters of a \(70 \%\)-alcohol solution?

A college fraternity house spent $$\$ 670$$ for an order of 85 pizzas. The order consisted of cheese pizzas, which cost $$\$ 5$$ each and Supreme pizzas, which cost $$\$ 12$$ each. Find the number of each kind of pizza ordered.

Part of $$\$ 8400$$ is invested at 5\%, and the remainder is invested at \(8 \%\). The total yearly interest from the two investments is $$\$ 576$$. Determine how much is invested at each rate.

If the numerator of a certain fraction is increased by 5 , and the denominator is decreased by 1 , the resulting fraction is \(\frac{8}{3}\). However, if the numerator of the original fraction is doubled, and the denominator is increased by 7 , then the resulting fraction is \(\frac{6}{11}\). Find the original fraction.

A man bought 2 pounds of coffee and 1 pound of butter for a total of $$\$9.25.$$ A month later, the prices had not changed (this makes it a fictitious problem), and he bought 3 pounds of coffee and 2 pounds of butter for $$\$ 15.50$. Find the price per pound of both the coffee and the butter.

Suppose that we have a rectangular-shaped book cover. If the width is increased by 2 centimeters, and the length is decreased by 1 centimeter, the area is increased by 28 square centimeters. However, if the width is decreased by 1 centimeter, and the length is increased by 2 centimeters, then the area is increased by 10 square centimeters. Find the dimensions of the book cover.

A blueprint indicates a master bedroom in the shape of a rectangle. If the width is increased by 2 feet and the length remains the same, then the area is increased by 36 square feet. However, if the width is increased by 1 foot and the length is increased by 2 feet, then the area is increased by 48 square feet. Find the dimensions of the room as indicated on the blueprint.

Give a general description of how to use the eliminationby-addition method to solve a system of two linear equations in two variables.

Explain how you would solve the system $$ \left(\begin{array}{lr} 3 x-4 y= & -1 \\ 2 x-5 y= & 9 \end{array}\right) $$ using the elimination-by-addition method.

There is another way of telling whether a system of two linear equations in two unknowns is consistent or inconsistent, or whether the equations are dependent, without taking the time to graph each equation. It can be shown that any system of the form $$ \begin{aligned} &a_{1} x+b_{1} y=c_{1} \\ &a_{2} x+b_{2} y=c_{2} \end{aligned} $$ has one and only one solution if $$ \frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}} $$ that it has no solution if $$ \frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}} $$ and that it has infinitely many solutions if $$ \frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}} $$ For each of the following systems, determine whether the system is consistent, the system is inconsistent, or the equations are dependent. (a) \(\left(\begin{array}{l}4 x-3 y=7 \\ 9 x+2 y=5\end{array}\right)\) (b) \(\left(\begin{array}{rl}5 x-y & =6 \\ 10 x-2 y & =19\end{array}\right)\) (c) \(\left(\begin{array}{l}5 x-4 y=11 \\ 4 x+5 y=12\end{array}\right)\) (d) \(\left(\begin{array}{l}x+2 y=5 \\ x-2 y=9\end{array}\right)\) (e) \(\left(\begin{array}{rl}x-3 y & =5 \\ 3 x-9 y & =15\end{array}\right)\) (f) \(\left(\begin{array}{rl}4 x+3 y & =7 \\ 2 x-y & =10\end{array}\right)\) (g) \(\left(\begin{array}{l}3 x+2 y=4 \\ y=-\frac{3}{2} x-1\end{array}\right)\) (h) \(\left(\begin{array}{l}y=\frac{4}{3} x-2 \\ 4 x-3 y=6\end{array}\right)\)

A system such as $$ \left(\begin{array}{l} \frac{3}{x}+\frac{2}{y}=2 \\ \frac{2}{x}-\frac{3}{y}=\frac{1}{4} \end{array}\right) $$ is not a system of linear equations but can be transformed into a linear system by changing variables. For example, when we substitute \(u\) for \(\frac{1}{x}\) and \(v\) for \(\frac{1}{y}\), the system cited becomes $$ \left(\begin{array}{l} 3 u+2 v=2 \\ 2 u-3 v=\frac{1}{4} \end{array}\right) $$ We can solve this "new" system either by elimination by addition or by substitution (we will leave the details for you) to produce \(u=\frac{1}{2}\) and \(v=\frac{1}{4}\). Therefore, because \(u=\frac{1}{x}\) and \(v=\frac{1}{y}\), we have \(\frac{1}{x}=\frac{1}{2} \quad\) and \(\quad \frac{1}{y}=\frac{1}{4}\) Solving these equations yields $$ x=2 \quad \text { and } \quad y=4 $$ The solution set of the original system is \(\\{(2,4)\\}\). Solve each of the following systems. (a) \(\left(\begin{array}{l}\frac{1}{x}+\frac{2}{y}=\frac{7}{12} \\\ \frac{3}{x}-\frac{2}{y}=\frac{5}{12}\end{array}\right)\) (b) \(\left(\begin{array}{r}\frac{2}{x}+\frac{3}{y}=\frac{19}{15} \\\ -\frac{2}{x}+\frac{1}{y}=-\frac{7}{15}\end{array}\right)\) (c) \(\left(\begin{array}{l}\frac{3}{x}-\frac{2}{y}=\frac{13}{6} \\\ \frac{2}{x}+\frac{3}{y}=0\end{array}\right)\) (d) \(\left(\begin{array}{l}\frac{4}{x}+\frac{1}{y}=11 \\\ \frac{3}{x}-\frac{5}{y}=-9\end{array}\right)\) (e) \(\left(\begin{array}{l}\frac{5}{x}-\frac{2}{y}=23 \\\ \frac{4}{x}+\frac{3}{y}=\frac{23}{2}\end{array}\right)\) (f) \(\left(\begin{array}{l}\frac{2}{x}-\frac{7}{y}=\frac{9}{10} \\\ \frac{5}{x}+\frac{4}{y}=-\frac{41}{20}\end{array}\right)\)