Systems of Equations and Inequalities

In Problems 37–72, solve each system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent. 

$\left\{\begin{array}{l}x+2y+z=1\\ 2x-y+2z=2\\ 3x+y+3z=3\end{array}\right.$

In Problems 37–72, solve each system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent.  

$\left\{\begin{array}{l}x+2y-z=3\\ 2x-y+2z=6\\ x-3y+3z=4\end{array}\right.$

In Problems 37–72, solve each system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent.  

$\left\{\begin{array}{l}\begin{array}{c}x-y+z=5\\ 3x+2y-2z=0\end{array}\end{array}\right.$

In Problems 37–72, solve each system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent.   

$\left\{\begin{array}{l}2x+y-z=4\\ -x+y+3z=1\end{array}\right.$

In Problems 37–72, solve each system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent.    

$\left\{\begin{array}{l}\begin{array}{c}2x+3y-z=3\\ x-y-z=0\end{array}\\ \begin{array}{c}-x+y+z=0\\ x+y+5z=5\end{array}\end{array}\right.$

In Problems 37–72, solve each system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent.     

$\left\{\begin{array}{l}\begin{array}{c}x-3y+z=1\\ 2x-y-4z=0\end{array}\\ \begin{array}{c}x-3y+2z=1\\ x-2y=5\end{array}\end{array}\right.$

In Problems 37–72, solve each system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent. 

$\left\{\begin{array}{l}4x+y+z-w=4\\ x-y+2z+3w=3\end{array}\right.$

In Problems 37–72, solve each system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent. 

$\left\{\begin{array}{c}-4x+y=5\\ 2x-y+z-w=5\\ z+w=4\end{array}\right.$

Curve Fitting Find the function $y=a{x}^2+bx+c$ whose graph contains the points $\left(1,2\right),\left(-2,-7\right)$ and $\left(2,-3\right)$.

Curve Fitting Find the function $y=a{x}^2+bx+c$ whose graph contains the points $\left(1,-1\right),\left(3,-1\right)$ and $\left(-2,14\right)$.

Curve Fitting Find the function $f\left(x\right) = a{x}^3 + b{x}^2 + cx + d$ for which $f(-3)=-112, f(-1)=-2, f\left(1\right)=4$, and $f\left(2\right)=13$. 

Curve Fitting Find the function $f\left(x\right)=a{x}^3+b{x}^2+cx+d$ for which $f(-2)=-10, f(-1)=3, f\left(1\right)=5$, and $f\left(3\right)=15$. 

Nutrition A dietitian at Palos Community Hospital wants a patient to have a meal that has 78 grams (g) of protein, 59 g of carbohydrates, and 75 milligrams (mg) of vitamin A. The hospital food service tells the dietitian that the dinner for today is salmon steak, baked eggs, and acorn squash. Each serving of salmon steak has 30 g of protein, 20 g of carbohydrates, and 2 mg of vitamin A. Each serving of baked eggs contains 15 g of protein, 2 g of carbohydrates, and 20 mg of vitamin A. Each serving of acorn squash contains 3 g of protein, 25 g of carbohydrates, and 32 mg of vitamin A. How many servings of each food should the dietitian provide for the patient? 

Nutrition A dietitian at General Hospital wants a patient to have a meal that has 47 grams (g) of protein, 58 g of carbohydrates, and 630 milligrams (mg) of calcium. The hospital food service tells the dietitian that the dinner for today is pork chops, corn on the cob, and 2% milk. Each serving of pork chops has 23 g of protein, 0 g of carbohydrates, and 10 mg of calcium. Each serving of corn on the cob contains 3 g of protein, 16 g of carbohydrates, and 10 mg of calcium. Each glass of 2% milk contains 9 g of protein, 13 g of carbohydrates, and 300 mg of calcium. How many servings of each food should the dietitian provide for the patient? 

Financial Planning Carletta has \(10,000 to invest. As her financial consultant, you recommend that she invest in Treasury bills that yield 6%, Treasury bonds that yield 7%, and corporate bonds that yield 8%. Carletta wants to have an annual income of \)680, and the amount invested in corporate bonds must be half that invested in Treasury bills. Find the amount in each investment. 

Landscaping A landscape company is hired to plant trees in three new subdivisions. The company charges the developer for each tree planted, an hourly rate to plant the trees, and a fixed delivery charge. In one subdivision it took 166 labor hours to plant 250 trees for a cost of \(7520. In a second subdivision it took 124 labor hours to plant 200 trees for a cost of \)5945. In the final subdivision it took 200 labor hours to plant 300 trees for a cost of $8985. Determine the cost for each tree, the hourly labor charge, and the fixed delivery charge.

Production To manufacture an automobile requires painting, drying, and polishing. Epsilon Motor Company produces three types of cars: the Delta, the Beta, and the sigma. Each Delta requires 10 hours (hr) for painting, 3 hr for drying, and 2 hr for polishing. A Beta requires 16 hr for painting, 5 hr for drying, and 3 hr for polishing, and a Sigma requires 8 hr for painting, 2 hr for drying, and 1 hr for polishing. If the company has 240 hr for painting, 69 hr for drying, and 41 hr for polishing per month, how many of each type of car are produced?

Production A Florida juice company completes the preparation of its products by sterilizing, filling, and labeling bottles. Each case of orange juice requires 9 minutes (min) for sterilizing, 6 min for filling, and 1 min for labeling. Each case of grapefruit juice requires 10 min for sterilizing, 4 min for filling, and 2 min for labeling. Each case of tomato juice requires 12 min for sterilizing, 4 min for filling, and 1 min for labeling. If the company runs the sterilizing machine for 398 min, the filling machine for 164 min, and the labeling machine for 58 min, how many cases of each type of juice are prepared?

Electricity: Kirchhoff’s Rules An application of Kirchhoff’s Rules to the circuit shown results in the following system of equations:

$\left\{\begin{array}{l}\begin{array}{c}-4+8-2I_2=0\\ 8=5I_4+I_1\\ 4=3I_3+I_1\end{array}\\ I_3+I_4=I_1 \end{array}\right.$

Find the currents $I_1, I_2, I_3, a\mathrm{nd} {\mathrm{I}}_4.$

Electricity: Kirchhoff’s Rules An application of Kirchhoff’s Rules to the circuit shown results in the following system of equations:

$\left\{\begin{array}{c}{I}_1=I_3+I_2\\ 24-6I_1-3I_3=0\\ 12+24-6I_1-6I_2=0\end{array}\right.$

Find the currents $I_1, I_{2,} \mathrm{and} I_3.$

Three retired couples each require an additional annual income of \(2000 per year. As their financial consultant, you recommend that they invest some money in Treasury bills that yield 7%, some money in corporate bonds that yield 9%, and some money in junk bonds that yield 11%. Prepare a table for each couple showing the various ways that their goals can be achieved:

(a) If the first couple has \)20,000 to invest.

(b) If the second couple has \(25,000 to invest.

(c) If the third couple has \)30,000 to invest.

(d) What advice would you give each couple regarding the

amount to invest and the choices available?

A young couple has \(25,000 to invest. As their financial consultant, you recommend that they invest some money in Treasury bills that yield 7%, some money in corporate bonds that yield 9%, and some money in junk bonds that yield 11%. Prepare a table showing the various ways that this couple can achieve the following goals:

(a) \)1500 per year in income

(b) \(2000 per year in income

(c) \)2500 per year in income

(d) What advice would you give this couple regarding the income that they require and the choices available?

A doctor’s prescription calls for a daily intake of a supplement containing 40 milligrams (mg) of vitamin C and 30 mg of vitamin D. Your pharmacy stocks three supplements that can be used: one contains 20% vitamin C and 30% vitamin D; a second, 40% vitamin C and 20% vitamin D; and a third, 30% vitamin C and 50% vitamin D. Create a table showing the possible combinations that could be used to fill the prescription.

A doctor’s prescription calls for the creation of pills that contain 12 units of vitamin $B_{12}$ and 12 units of vitamin E. Your pharmacy stocks three powders that can be used to make these pills: one contains 20% vitamin B12 and 30% vitamin E; a second, 40% vitamin $B_{12}$ and 20% vitamin E; and a third, 30% vitamin $B_{12}$ and 40% vitamin E. Create a table showing the possible combinations of each powder that could be mixed in each pill.

Write a brief paragraph or two that outlines your strategy for solving a system of linear equations using matrices.

When solving a system of linear equations using matrices, do you prefer to place the augmented matrix in row echelon form or in reduced row echelon form? Give reasons for your choice.

Make up a system of three linear equations containing three variables that has:

Part (a): No solution

Part (b): Exactly one solution

Part (c): Infinitely many solutions

In Problems 15 – 42, solve each system of equations using Cramer’s Rule if it is applicable. If Cramer’s Rule is not applicable, say so.

$\left\{\begin{array}{l}3x-5y=3\\ 15x+5y=21\end{array}\right.$

$D=\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|=\_\_\_\_\_\_\_\_$

Using Cramer’s Rule, the value of x that satisfies the system of equations $\left\{\begin{array}{l}2x+3y=5\\ x-4y=-3\end{array}\right.$  is $x=\frac{}{\left|\begin{array}{cc}2& 3\\ 1& -4\end{array}\right|}$.

True or False A determinant can never equal 0. 

True or False When using Cramer’s Rule, if D = 0, then the system of linear equations is inconsistent. 

True or False The value of a determinant remains unchanged if any two rows or any two columns are interchanged. 

True or False If any row (or any column) of a determinant is multiplied by a nonzero number k, the value of the determinant remains unchanged. 

Find the value of the determinant.

$\left|\begin{array}{cc}6& 4\\ -1& 3\end{array}\right|$

Find the value of the determinant.

$\left|\begin{array}{cc}8& -3\\ 4& 2\end{array}\right|$

Find the value of the determinant.

$\left|\begin{array}{cc}-3& -1\\ 4& 2\end{array}\right|$

Find the value of the determinant.

$\left|\begin{array}{cc}-4& 2\\ -5& 3\end{array}\right|$

Find the value of the determinant.

$\left|\begin{array}{ccc}3& 4& 2\\ 1& -1& 5\\ 1& 2& -2\end{array}\right|$

Find the value of the determinant.

$\left|\begin{array}{ccc}1& 3& -2\\ 6& 1& -5\\ 8& 2& 3\end{array}\right|$

Find the value of the determinant.

$\left|\begin{array}{ccc}4& -1& 2\\ 6& -1& 0\\ 1& -3& 4\end{array}\right|$

Find the value of the determinant.

$\left|\begin{array}{ccc}3& -9& 4\\ 1& 4& 0\\ 8& -3& 1\end{array}\right|$

Solve the system of equations using Cramer’s Rule if it is applicable. If Cramer’s Rule is not applicable, say so.

$\left\{\begin{array}{l}x+y=8\\ x-y=4\end{array}\right.$

In Problems 15 – 42, solve each system of equations using Cramer’s Rule if it is applicable. If Cramer’s Rule is not applicable, say so.

$\left\{\begin{array}{l}x+2y=5\\ x-y=3\end{array}\right.$

In Problems 15 – 42, solve each system of equations using Cramer’s Rule if it is applicable. If Cramer’s Rule is not applicable, say so. 

$\left\{\begin{array}{l}5x-y=13\\ 2x+3y=12\end{array}\right.$

In Problems 15 – 42, solve each system of equations using Cramer’s Rule if it is applicable. If Cramer’s Rule is not applicable, say so. 

$\left\{\begin{array}{l}x+3y=5\\ 2x-3y=-8\end{array}\right.$

In Problems 15 – 42, solve each system of equations using Cramer’s Rule if it is applicable. If Cramer’s Rule is not applicable, say so.  

$\left\{\begin{array}{l}3x=24\\ x+2y=0\end{array}\right.$

In Problems 15 – 42, solve each system of equations using Cramer’s Rule if it is applicable. If Cramer’s Rule is not applicable, say so. 

$\left\{\begin{array}{l}4x+5y=-3\\ -2y=-4\end{array}\right.$

In Problems 15 – 42, solve each system of equations using Cramer’s Rule if it is applicable. If Cramer’s Rule is not applicable, say so. 

$\left\{\begin{array}{l}3x-6y=24\\ 5x+4y=12\end{array}\right.$

In Problems 15 – 42, solve each system of equations using Cramer’s Rule if it is applicable. If Cramer’s Rule is not applicable, say so. 

$\left\{\begin{array}{l}2x+4y=16\\ 3x-5y=-9\end{array}\right.$