Question

properties objective e

1. determine whether each expression is a linear combination.

a. ( 3x + 4y )________ b. ( 8.3k - 9.2j )________
c. ( 2x^{2}-3x )________ d. ( 6a + 5b - 2c )________
in 2 and 3, a. write a linear combination representing the total cost, and
b. determine whether the situation described is discrete or continuous.

1. apples cost ( $0.75 ) each and bananas cost ( $0.45 ) each.

a. ______________________ b. ______________________

1. apples cost ( $1.29 ) per pound and bananas cost ( $0.79 ) per pound.

a. ______________________ b. ______________________
uses objective h

1. at a party store, helium balloons cost ( 79 ) each and a box of favors costs( $11.95 ). you have ( $40 ) to spend and you need two boxes of favors. how

many balloons can you buy? there is no sales tax.

1. a chemist has solutions of hydrochloric acid in two different

concentrations. one solution has 10 moles per liter; the other has
2.5 moles per liter. she mixes ( x ) liters of the first solution and ( y ) liters
of the second solution.
a. how many moles of acid are in ( x ) liters of the first solution? __________
b. how many moles of acid are in ( y ) liters of the second solution? __________
c. how many moles of acid are in the mixture? __________
d. the chemist wants to have 6 moles of acid in the final mixture. she uses 1 liter of the se
solution. how many liters of the first solution should she use?
__________

1. describe a situation leading to the equation ( 3x + 5y = 45 ).

Explanation:

Step1: Determine linear combination for 1a

A linear combination has variables with degree 1. In \(3x + 4y\), \(x\) and \(y\) have degree 1.

Step2: Determine linear combination for 1b

In \(8.3k-9.2j\), \(k\) and \(j\) have degree 1.

Step3: Determine linear combination for 1c

In \(2x^{2}-3x\), \(x\) has degree 2 (from \(x^{2}\)), so it is not a linear combination.

Step4: Determine linear combination for 1d

In \(6a + 5b-2c\), \(a\), \(b\), and \(c\) have degree 1.

Step5: Solve problem 2a

Let \(x\) be the number of apples and \(y\) be the number of bananas. The total cost \(C=0.75x + 0.45y\).

Step6: Solve problem 2b

The number of apples and bananas (\(x,y\)) can only be whole numbers (0,1,2,...), so the situation is discrete.

Step7: Solve problem 3a

Let \(x\) be the pounds of apples and \(y\) be the pounds of bananas. The total cost \(C = 1.29x+0.79y\).

Step8: Solve problem 3b

Pounds (\(x,y\)) can be any non - negative real number (e.g., \(x = 1.5\) pounds), so the situation is continuous.

Step9: Solve problem 4

Cost of two boxes of favors: \(2\times11.95=\$23.9\). Money left for balloons: \(40 - 23.9=\$16.1\). Number of balloons \(n=\frac{16.1}{0.79}\approx20.38\). Since we can't buy a fraction of a balloon, \(n = 20\).

Step10: Solve problem 5a

Moles in \(x\) liters of first solution: \(10x\) (using \(moles=\text{concentration}\times\text{volume}\)).

Step11: Solve problem 5b

Moles in \(y\) liters of second solution: \(2.5y\).

Step12: Solve problem 5c

Moles in the mixture: \(10x + 2.5y\).

Step13: Solve problem 5d

Given \(y = 1\) and \(10x+2.5y=6\). Substitute \(y = 1\) into the equation: \(10x+2.5\times1=6\). Then \(10x=6 - 2.5=3.5\), so \(x=\frac{3.5}{10}=0.35\) liters.

Answer:

1. a. Yes; b. Yes; c. No; d. Yes
2. a. \(C = 0.75x+0.45y\); b. Discrete
3. a. \(C = 1.29x + 0.79y\); b. Continuous
4. 20
5. a. \(10x\); b. \(2.5y\); c. \(10x + 2.5y\); d. \(0.35\) liters
6. (For example) Let \(x\) be the number of pens that cost \(\$3\) each and \(y\) be the number of notebooks that cost \(\$5\) each. The total money spent is \(\$45\), so \(3x + 5y=45\).