1. Use the echelon method to solve the following system of two equations in two unknowns. Check your answer.
\[
\begin{align*}
4x - 3y &= -1 \\
-8x + 6y &= 2
\end{align*}
\]
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The solution is (Type an ordered pair.)
B. There are infinitely many solutions. The solution is \(y\), where \(y\) is any real number.
C. There is no solution.
2. Use the echelon method to solve the given system of two equations in two unknowns. Check your answers.
\[
\begin{align*}
x + y &= -143 \\
5x + y &= -285
\end{align*}
\]
Select the correct choice below and fill in any answer boxes in your choice.
A. The solution of the system is (Simplify your answer. Type an ordered pair.)
B. There are infinitely many solutions. The solution is \(y\), where \(y\) is any real number.
C. There is no solution.
3. For the following system of equations in echelon form, tell how many solutions there are in nonnegative integers.
\[
\begin{align*}
x + 2y + 3z &= 80 \\
3y + 4z &= 24
\end{align*}
\]
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. There are nonnegative solutions.
B. There are infinitely many solutions.
C. There is no solution.
4. For the following system of equations in echelon form, tell how many solutions there are in nonnegative integers.
\[
\begin{align*}
2x + 2y + 4z &= 100 \\
y - 4z &= 30
\end{align*}
\]
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. There are nonnegative solutions.
B. There are infinitely many solutions.
C. There is no solution.
5. If 30 lb of rice and 20 lb of potatoes cost $15.20, and 20 lb of rice and 12 lb of potatoes cost $9.52, how much will 10 lb of rice and 50 lb of potatoes cost?
Let \(x\) be the cost of 1 lb of rice and \(y\) be the cost of 1 lb of potatoes. Set up two linear equations from the given information using \(x\) and \(y\) as the variables.
6. Lorri Morgan has $17,000 invested in company A and company B stock. The company A stock currently sells for $90 a share and the company B stock for $72 a share. Her stockbroker points out that if company A stock goes up 50% and company B stock goes up by $36 a share, her stock will be worth $24,500. Is this situation possible?
Yes or No
7. A furniture company makes dining room furniture. A buffet requires 30 hours for construction and 10 hours for finishing. A chair requires 20 hours for construction and 20 hours for finishing. A table requires 20 hours for construction and 60 hours for finishing. The construction department has 420 hours of labor and the finishing department has 220 hours of labor available each week. How many pieces of each type of furniture should be produced each week if the factory is to run at full capacity?
A. The number of pieces of each type of furniture that should be produced each week cannot be found.
B. Either 10 buffets, 6 chairs, and no tables or 12 buffets, 2 chairs, and 1 table
C. Either 1 buffet, 6 chairs, and 10 tables or no buffets, 12 chairs, and 1 table
D. 6 buffets, 10 chairs, and 1 table
8. The date of the first sighting of robins has been occurring earlier each spring over the past 25 years at a certain laboratory. Scientists from this laboratory have developed two linear equations, shown below, that estimate the date of the first sighting of robins, where \(x\) is the year and \(y\) is the estimated number of days into the year when a robin can be expected. Complete parts a and b. (Hint: 2000 was a leap year.)
\[
\begin{align*}
y &= 862 - 0.396x \\
y &= 1596 - 0.756x
\end{align*}
\]
a. Compare the date of first sighting in 2000 for each of these equations.
- March 11, March 25
- March 7, March 6
- March 10, March 24
- March 18, March 10
9. The object of a number game is to combine four numbers, using addition, subtraction, multiplication, and/or division, to get the number 24. For example, the numbers 2, 5, 5, 4 can be combined as \(2(5 + 5) + 4 = 24\). For the algebra edition of the game and the game card shown to the right, the object is to find single-digit positive integer values \(x\) and \(y\) so the found numbers \(x+y\), \(3x+2y\), 8, and 9 can be combined to make 24. Use this information to answer parts (a) and (b).
(a) Using the game card, write a system of equations that, when solved, can be used to make 24 from the game card. What is the solution to this system and how can it be used to make 24 on the game card?
What is a possible system?
A.
\[
\begin{align*}
x + y &= 3 \\
3x + 2y &= 8
\end{align*}
\]
B.
\[
\begin{align*}
x + y &= 1 \\
3x + 2y &= 2
\end{align*}
\]
C.
\[
\begin{align*}
x + y &= 5 \\
3x + 2y &= 9
\end{align*}
\]
D.
\[
\begin{align*}
x + y &= 4 \\
3x + 2y &= 5
\end{align*}
\]
