Chapter 10

For Problems \(17-32\), indicate the solution set for each system of inequalities by shading the appropriate region. $$ \left(\begin{array}{l} x<3 \\ y>2 \end{array}\right) $$

What happens if you try to graph the system $$ \left(\begin{array}{rr} x^{2}+4 y^{2}= & 16 \\ 2 x^{2}+5 y^{2}= & -12 \end{array}\right) $$

For Problems \(11-30\), use Cramer's rule to find the solution set of each system. (Objective 2) $$ \left(\begin{array}{rr} 5 x-y+2 z= & 10 \\ 7 x+2 y-2 z= & -4 \\ -3 x-y+4 z= & 1 \end{array}\right) $$

Use Cramer's rule to find the solution set for each of the following systems. (Objective 2) $$ \left(\begin{array}{rr} 6 x-y= & 0 \\ 5 x+4 y= & 29 \end{array}\right) $$

Part of $$\$ 3000$$ is invested at \(4 \%\), another part at \(5 \%\), and the remainder at \(6 \%\). The total yearly income from the three investments is $$\$ 160$$. The sum of the amounts invested at \(4 \%\) and \(5 \%\) equals the amount invested at \(6 \%\). Determine how much is invested at each rate.

For Problems 19-48, solve each system by using either the substitution or the elimination-by-addition method, whichever seems more appropriate. (Objective 2) $$ \left(\begin{array}{l} 3 x-2 y=7 \\ 5 x+7 y=1 \end{array}\right) $$

Two numbers are added together, and the sum is \(131 .\) One number is five less than three times the other. Find the two numbers.

For Problems \(17-32\), indicate the solution set for each system of inequalities by shading the appropriate region. $$ \left(\begin{array}{l} 2 x-y>4 \\ 2 x-y>0 \end{array}\right) $$

Explain how you would solve the system $$ \left(\begin{array}{l} x^{2}+y^{2}=9 \\ y^{2}=x^{2}+4 \end{array}\right) $$

For Problems \(11-30\), use Cramer's rule to find the solution set of each system. (Objective 2) $$ \left(\begin{array}{rl} 4 x-y-3 z= & -12 \\ 5 x+y+6 z= & 4 \\ 6 x-y-3 z= & -14 \end{array}\right) $$

Use Cramer's rule to find the solution set for each of the following systems. (Objective 2) $$ \left(\begin{array}{l} 3 x-4 y=2 \\ 9 x-12 y=6 \end{array}\right) $$

The perimeter of a triangle is 45 centimeters. The longest side is 4 centimeters less than twice the shortest side. The sum of the lengths of the shortest and longest sides is 7 centimeters less than three times the length of the remaining side. Find the lengths of all three sides of the triangle.

For Problems 19-48, solve each system by using either the substitution or the elimination-by-addition method, whichever seems more appropriate. (Objective 2) $$ \left(\begin{array}{l} 4 x+7 y=2 \\ 9 x-2 y=1 \end{array}\right) $$

The length of a rectangle is twice the width of the rectangle. Given that the perimeter of the rectangle is 72 centimeters, find the dimensions.

Use Cramer's rule to find the solution set for each of the following systems. (Objective 2) $$ \left(\begin{array}{rr} -4 x+3 y & =3 \\ 4 x-6 y & =-5 \end{array}\right) $$

Give a step-by-step description of how to solve the system $$ \left(\begin{array}{rr} x-2 y+3 z= & -23 \\ 5 y-2 z= & 32 \\ 4 z= & -24 \end{array}\right) $$

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

Two angles are complementary, and the measure of one of the angles is \(10^{\circ}\) less than four times the measure of the other angle. Find the measure of each angle.

For Problems \(17-32\), indicate the solution set for each system of inequalities by shading the appropriate region. $$ \left(\begin{array}{l} 3 x-2 y<6 \\ 2 x-3 y<6 \end{array}\right) $$

For each of the following systems, (a) use your graphing calculator to show that there are no real number solutions, and (b) solve the system by the substitution method or the elimination-by-addition method to find the complex solutions. (a) \(\left(\begin{array}{l}y=x^{2}+1 \\ y=-3\end{array}\right)\) (b) \(\left(\begin{array}{l}y=-x^{2}+1 \\ y=3\end{array}\right)\) (c) \(\left(\begin{array}{r}y=x^{2} \\ x-y=4\end{array}\right)\) (d) \(\left(\begin{array}{l}y=x^{2}+1 \\ y=-x^{2}\end{array}\right)\) (e) \(\left(\begin{array}{l}x^{2}+y^{2}=1 \\ x+y=2\end{array}\right)\) (f) \(\left(\begin{array}{l}x^{2}+y^{2}=2 \\ x^{2}-y^{2}=6\end{array}\right)\)

Explain how to use determinants to solve the system $$ \left(\begin{array}{rr} x-2 y+z= & 1 \\ 2 x-y-z= & 5 \\ 5 x+3 y+4 z= & -6 \end{array}\right) $$

Use Cramer's rule to find the solution set for each of the following systems. (Objective 2) $$ \left(\begin{array}{l} x-2 y=-1 \\ x=-6 y+5 \end{array}\right) $$

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

The difference of two numbers is 75 . The larger number is three less than four times the smaller number. Find the numbers.

For Problems \(17-32\), indicate the solution set for each system of inequalities by shading the appropriate region. $$ \left(\begin{array}{l} 2 x+5 y>10 \\ 5 x+2 y>10 \end{array}\right) $$

Graph the system \(\left(\begin{array}{l}y=x^{2}+2 \\ 6 x-4 y=-5\end{array}\right)\) and use the TRACE and ZOOM features of your calculator to demonstrate clearly that this system has no real number solutions.

Evaluate the following determinant by expanding about the second column. $$ \left|\begin{array}{lll} a & e & a \\ b & f & b \\ c & g & c \end{array}\right| $$ Make a conjecture about determinants that contain two identical columns.

Use Cramer's rule to find the solution set for each of the following systems. (Objective 2) $$ \left(\begin{array}{l} 6 x-5 y=1 \\ 4 x+7 y=2 \end{array}\right) $$

For Problems 19-48, solve each system by using either the substitution or the elimination-by-addition method, whichever seems more appropriate. (Objective 2) $$ \left(\begin{array}{l} -2 x+5 y=-16 \\ x=\frac{3}{4} y+1 \end{array}\right) $$

In a class of 50 students, the number of females is two more than five times the number of males. How many females are there in the class?

How do you know by inspection, without graphing, that the solution set of the system \(\left(\begin{array}{l}3 x-2 y>5 \\ 3 x-2 y<2\end{array}\right)\) is the null set?

Show that \(\left|\begin{array}{rrr}1 & -1 & 2 \\ 2 & 3 & -1 \\ -1 & 2 & 4\end{array}\right|=-\left|\begin{array}{rrr}-1 & 1 & 2 \\ 3 & 2 & -1 \\ 2 & -1 & 4\end{array}\right|\). Make a conjecture about the result of interchanging two columns of 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) $$ \left(\begin{array}{c} y=\frac{2}{3} x-\frac{3}{4} \\ 2 x+3 y=11 \end{array}\right) $$

In a recent survey, one thousand registered voters were asked about their political preferences. The number of males in the survey was five less than one-half of the number of females. Find the number of males in the survey.

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

(a) Show that \(\left|\begin{array}{rrr}2 & 1 & 2 \\ 4 & -1 & -2 \\ 6 & 3 & 1\end{array}\right|=2\left|\begin{array}{rrr}1 & 1 & 2 \\ 2 & -1 & -2 \\ 3 & 3 & 1\end{array}\right|\). Make a conjecture about the result of factoring a common factor from each element of a column in a determinant. (b) Use your conjecture from part (a) to help evaluate the following determinant. $$ \left|\begin{array}{rrr} 2 & 4 & -1 \\ -3 & -4 & -2 \\ 5 & 4 & 3 \end{array}\right| $$

Use Cramer's rule to find the solution set for each of the following systems. (Objective 2) $$ \left(\begin{array}{l} 7 x+2 y=-1 \\ y=-x+2 \end{array}\right) $$

For Problems 19-48, solve each system by using either the substitution or the elimination-by-addition method, whichever seems more appropriate. (Objective 2) $$ \left(\begin{array}{l} y=\frac{2}{3} x-4 \\ 5 x-3 y=9 \end{array}\right) $$

The perimeter of a rectangle is 94 inches. The length of the rectangle is 7 inches more than the width. Find the dimensions of the rectangle.

Use Cramer's rule to find the solution set for each of the following systems. (Objective 2) $$ \left(\begin{array}{l} 9 x-y=-2 \\ y=4-8 x \end{array}\right) $$

For Problems 19-48, solve each system by using either the substitution or the elimination-by-addition method, whichever seems more appropriate. (Objective 2) $$ \left(\begin{array}{c} 5 x-3 y=7 \\ x=\frac{3 y}{4}-\frac{1}{3} \end{array}\right) $$

Two angles are supplementary, and the measure of one of them is \(20^{\circ}\) less than three times the measure of the other angle. Find the measure of each angle.

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) \(\left(\begin{array}{l}3.7 x-2.9 y=-14.3 \\ 1.6 x+4.7 y=-30\end{array}\right)\)

Use Cramer's rule to find the solution set for each of the following systems. (Objective 2) $$ \left(\begin{array}{c} -\frac{2}{3} x+\frac{1}{2} y=-7 \\ \frac{1}{3} x-\frac{3}{2} y=6 \end{array}\right) $$

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

A deposit slip listed $$\$ 700$$ in cash to be deposited. There were 100 bills, some of them five-dollar bills and the remainder ten-dollar bills. How many bills of each denomination were deposited?

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

For Problems 19-48, solve each system by using either the substitution or the elimination-by-addition method, whichever seems more appropriate. (Objective 2) $$ \left(\begin{array}{l} \frac{3 x}{4}-\frac{2 y}{3}=31 \\ \frac{7 x}{5}+\frac{y}{4}=22 \end{array}\right) $$

Cindy has 30 coins, consisting of dimes and quarters, that total $$\$ 5.10$$. How many coins of each kind does she have?

Use Cramer's rule to find the solution set for each of the following systems. (Objective 2) $$ \left(\begin{array}{rl} x+\frac{2}{3} y & =-6 \\ -\frac{1}{4} x+3 y & =-8 \end{array}\right) $$