Chapter 10

Use Cramer's rule to find the solution set for each of the following systems. (Objective 2) $$ \left(\begin{array}{l} 4 x-y=11 \\ 2 x+3 y=23 \end{array}\right) $$

For Problems \(1-22\), solve each of the systems and use matrices as we did in the examples of this section. $$ \left(\begin{array}{rr} 2 x+5 y+z= & 1 \\ x+2 y-3 z= & -13 \\ 3 x-y-2 z= & -4 \end{array}\right) $$

The sum of the digits of a three-digit number is 13 . The sum of the hundreds digit and the tens digit is 1 less than the units digit. The sum of three times the hundreds digit and four times the units digit is 26 more than twice the tens digit. Find the number.

For Problems 19-48, solve each system by using either the substitution or the elimination-by-addition method, whichever seems more appropriate. (Objective 2) $$ \left(\begin{array}{l} 11 x-3 y=-60 \\ y=-38-6 x \end{array}\right) $$

For Problems \(1-26\), solve each system by using the substitution method. (Objective 1) $$ \left(\begin{array}{l} 2 x-3 y=-16 \\ 6 x+7 y=16 \end{array}\right) $$

For Problems \(17-32\), indicate the solution set for each system of inequalities by shading the appropriate region. $$ \left(\begin{array}{l} x+y>1 \\ x-y<1 \end{array}\right) $$

For Problems \(1-28\), (a) graph each system so that approximate real number solutions (if there are any) can be predicted, and (b) solve each system using the substitution method or the elimination-by-addition method. (Objectives 1 and 2) $$ \left(\begin{array}{l} y=-x^{2}-3 \\ y=-2 x^{2}+1 \end{array}\right) $$

Use Cramer's rule to find the solution set for each of the following systems. (Objective 2) $$ \left(\begin{array}{rr} -x+3 y= & 17 \\ 4 x-5 y= & -33 \end{array}\right) $$

What is a matrix? What is an augmented matrix of a system of linear equations?

Two bottles of catsup, 2 jars of peanut butter, and 1 jar of pickles cost $$\$ 7.78$$. Three bottles of catsup, 4 jars of peanut butter, and 2 jars of pickles cost $$\$ 14.34$$. Four bottles of catsup, 3 jars of peanut butter, and 5 jars of pickles cost $$\$ 19.19$$. Find the cost per bottle of catsup, the cost per jar of peanut butter, and the cost per jar of pickles.

For Problems \(1-26\), solve each system by using the substitution method. (Objective 1) $$ \left(\begin{array}{rr} 4 x-5 y= & 3 \\ 8 x+15 y= & -24 \end{array}\right) $$

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

For Problems \(1-28\), (a) graph each system so that approximate real number solutions (if there are any) can be predicted, and (b) solve each system using the substitution method or the elimination-by-addition method. (Objectives 1 and 2) $$ \left(\begin{array}{l} y=x^{2}+2 \\ y=2 x^{2}+1 \end{array}\right) $$

For Problems \(11-30\), use Cramer's rule to find the solution set of each system. (Objective 2) $$ \left(\begin{array}{rr} 6 x-5 y+2 z= & 7 \\ 2 x+3 y-4 z= & -21 \\ 2 y+3 z= & 10 \end{array}\right) $$

Use Cramer's rule to find the solution set for each of the following systems. (Objective 2) $$ \left(\begin{array}{rr} 5 x+2 y= & -15 \\ 7 x-3 y= & 37 \end{array}\right) $$

Describe how to use matrices to solve the system $$ \left(\begin{array}{r} x-2 y=5 \\ 2 x+7 y=9 \end{array}\right) $$

Five pounds of potatoes, 1 pound of onions, and 2 pounds of apples cost $$\$ 3.80$$. Two pounds of potatoes, 3 pounds of onions, and 4 pounds of apples cost $$\$ 5.78$$. Three pounds of potatoes, 4 pounds of onions, and 1 pound of apples cost $$\$ 4.08$$. Find the price per pound for each item.

For Problems 19-48, solve each system by using either the substitution or the elimination-by-addition method, whichever seems more appropriate. (Objective 2) $$ \left(\begin{array}{l} y=3 x+34 \\ y=-8 x-54 \end{array}\right) $$

For Problems \(1-26\), solve each system by using the substitution method. (Objective 1) $$ \left(\begin{array}{l} 2 x+3 y=3 \\ 4 x-9 y=-4 \end{array}\right) $$

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

For Problems \(1-28\), (a) graph each system so that approximate real number solutions (if there are any) can be predicted, and (b) solve each system using the substitution method or the elimination-by-addition method. (Objectives 1 and 2) $$ \left(\begin{array}{l} x^{2}+y^{2}=2 \\ x-y=4 \end{array}\right) $$

For Problems \(11-30\), use Cramer's rule to find the solution set of each system. (Objective 2) $$ \left(\begin{array}{rl} -2 x+5 y-3 z & =-1 \\ 2 x-7 y+3 z & =1 \\ 4 x-y-6 z & =-6 \end{array}\right) $$

Use Cramer's rule to find the solution set for each of the following systems. (Objective 2) $$ \left(\begin{array}{l} 9 x+5 y=-8 \\ 7 x-4 y=-22 \end{array}\right) $$

Solve the system $$ \left(\begin{array}{r} x-3 y-2 z+w=-3 \\ -2 x+7 y+z-2 w=-1 \\ 3 x-7 y-3 z+3 w=-5 \\ 5 x+y+4 z-2 w=18 \end{array}\right) $$

The sum of three numbers is 20 . The sum of the first and third numbers is 2 more than twice the second number. The third number minus the first yields three times the second number. Find the numbers.

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

For Problems \(1-26\), solve each system by using the substitution method. (Objective 1) $$ \left(\begin{array}{l} 6 x-3 y=4 \\ 5 x+2 y=-1 \end{array}\right) $$

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

For Problems \(1-28\), (a) graph each system so that approximate real number solutions (if there are any) can be predicted, and (b) solve each system using the substitution method or the elimination-by-addition method. (Objectives 1 and 2) $$ \left(\begin{array}{l} y=-x^{2}+1 \\ x+y=2 \end{array}\right) $$

For Problems \(11-30\), use Cramer's rule to find the solution set of each system. (Objective 2) $$ \left(\begin{array}{rr} 7 x-2 y+3 z= & -4 \\ 5 x+2 y-3 z= & 4 \\ -3 x-6 y+12 z= & -13 \end{array}\right) $$

Use Cramer's rule to find the solution set for each of the following systems. (Objective 2) $$ \left(\begin{array}{rr} 8 x-11 y= & 3 \\ -x+4 y= & -3 \end{array}\right) $$

Solve the system $$ \left(\begin{array}{rr} x-2 y+2 z-w= & -2 \\ -3 x+5 y-z-3 w= & 2 \\ 2 x+3 y+3 z+5 w= & -9 \\ 4 x-y-z-2 w= & 8 \end{array}\right) $$

The sum of three numbers is 40 . The third number is 10 less than the sum of the first two numbers. The second number is 1 larger than the first. Find the numbers.

For Problems 19-48, solve each system by using either the substitution or the elimination-by-addition method, whichever seems more appropriate. (Objective 2) $$ \left(\begin{array}{l} 3 x-y=9 \\ 5 x+7 y=1 \end{array}\right) $$

For Problems \(1-26\), solve each system by using the substitution method. (Objective 1) $$ \left(\begin{array}{l} 7 x-2 y=1 \\ 4 x+5 y=2 \end{array}\right) $$

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

For Problems \(1-28\), (a) graph each system so that approximate real number solutions (if there are any) can be predicted, and (b) solve each system using the substitution method or the elimination-by-addition method. (Objectives 1 and 2) $$ \left(\begin{array}{l} 2 x+y=6 \\ x^{2}+y^{2}=4 \end{array}\right) $$

For Problems \(11-30\), use Cramer's rule to find the solution set of each system. (Objective 2) $$ \left(\begin{array}{rr} -x-y+5 z= & 4 \\ x+y-7 z= & -6 \\ 2 x+3 y+4 z= & 13 \end{array}\right) $$

Suppose that the augmented matrix of a system of three equations in three variables can be changed to the following matrix. $$ \left[\begin{array}{rrr:r} 1 & 1 & -2 & 4 \\ 0 & -5 & 11 & -13 \\ 0 & 0 & 0 & -9 \end{array}\right] $$ What can be said about the solution set of the system?

The sum of the measures of the angles of a triangle is \(180^{\circ}\). The largest angle is twice the smallest angle. The sum of the smallest and the largest angle is twice the other angle. Find the measure of each angle.

For Problems 19-48, solve each system by using either the substitution or the elimination-by-addition method, whichever seems more appropriate. (Objective 2) $$ \left(\begin{array}{r} 5 x-2 y=1 \\ 10 x-4 y=7 \end{array}\right) $$

For Problems \(27-40\), solve each problem by setting up and solving an appropriate system of equations. (Objective 2 ) Doris invested some money at \(7 \%\) and some money at \(8 \%\). She invested \(\$ 6000\) more at \(8 \%\) than she did at \(7 \%\). Her total yearly interest from the two investments was \(\$ 780\). How much did Doris invest at each rate?

For Problems \(17-32\), indicate the solution set for each system of inequalities by shading the appropriate region. $$ \left(\begin{array}{l} x \geq-1 \\ y<4 \end{array}\right) $$

For Problems \(1-28\), (a) graph each system so that approximate real number solutions (if there are any) can be predicted, and (b) solve each system using the substitution method or the elimination-by-addition method. (Objectives 1 and 2) $$ \left(\begin{array}{l} y=x^{2}-1 \\ x-y=3 \end{array}\right) $$

For Problems \(11-30\), use Cramer's rule to find the solution set of each system. (Objective 2) $$ \left(\begin{array}{rr} x+7 y-z= & -1 \\ -x-9 y+z= & 3 \\ 3 x+4 y-6 z= & 5 \end{array}\right) $$

Use Cramer's rule to find the solution set for each of the following systems. (Objective 2) $$ \left(\begin{array}{l} 4 x-7 y=0 \\ 7 x+2 y=0 \end{array}\right) $$

Suppose that the augmented matrix of a system of three linear equations in three variables can be changed to the following matrix. $$ \left[\begin{array}{rrr:r} 1 & 0 & 1 & 1 \\ 0 & 1 & -1 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] $$ What can be said about the solution set of the system?

A box contains $$\$ 2$$ in nickels, dimes, and quarters. There are 19 coins in all, and there are twice as many nickels as dimes. How many coins of each kind are there?

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

Suppose that Gus invested a total of $$\$ 8000$$, part of it at \(8 \%\) and the remainder at \(9 \%\). His yearly income from the two investments was $$\$ 690$$. How much did he invest at each rate?