Math 116 Algebra 1a wk 7 ch. 8

Topic: Systems of Equations
Refer to these examples:
Ex. 2: EXAMPLE 2 Blending Flower Seeds. Tara’s website, Garden Edibles, specializes in the sale of herbs and flowers for colorful meals and garnishes. Tara sells packets of nasturtium seeds for $0.95 each and packets of Johnny-jumpup seeds for $1.43 each. She decides to offer a 16-packet spring-garden combination, combining packets of both types of seeds at $1.10 per packet. How many packets of each type of seed should be put in her garden mix?

|Let the garden mix have x packets of Nasturtium seeds and y packets of Johnny-jumpup seeds. |
|Then, by data we have |
|x + y = 16 ( y = 16 – x |
|0.95x + 1.43y = 16(1.10) |
|0.95x + 1.43(16 – x) = 17.6 |
|0.95x – 1.43x = 17.6 – 1.43(16) |
|-0.48x = -5.28 |
|x = -5.28/-0.48 = 11 packets (Nasturtium seeds) and y = 16 – 11 = 5 packets (Johnny-jumpup seeds) |

EXAMPLE 3 Student Loans. Jed’s student loans totaled $16,200. Part was a Perkins loan made at 5% interest and the rest was a Stafford loan made at 4% interest. After one year, Jed’s loans accumulated $715 in interest. What was the amount of each loan?

|Let the amount of Perkin’s loan be x and that of Stafford loan be y. |
|Then, by data we have |
|x + y = 16200 ( y = 16200 – x |
|0.05x + 0.04y = 715 |
|0.05x + 0.04(16200 – x) = 715 |
|0.05x – 0.04x = 715 – 0.04(16200) |
|0.01x = 67 |
|x = 67/0.01 = $6700 (Perkin’s loan) and y = 16200 – 6700 = $9500 (Stafford loan) |

EXAMPLE 4 Mixing Fertilizers. Yardbird Gardening carries two kinds of fertilizer containing nitrogen and water. “Gently Green” is 5% nitrogen and
“Sun Saver” is 15% nitrogen. Yardbird Gardening needs to combine the two types of solution to make 90 L of a solution that is 12% nitrogen. How much of each brand should be used?

|Let x L of Gently Green and y L of Sun Saver be required for the mix. |
|Then, by data we have |
|x + y = 90 ( y = 90 – x |
|0.05x + 0.15y = 90(0.12) |
|0.05x + 0.15(90 – x) = 10.8 |
|0.05x – 0.15x = 10.8 – 0.15(90) |
|-0.1x = -2.7 |
|x = -2.7/-0.1 = 27 L (Gently Green) and 90 – 27 = 63 L (Sun Saver). |

DQ 2 wk 7 Review examples 2, 3, and 4 in section 8.4 of the text. How does the author determine what the first equation should be? What about the second equation? How are these examples similar? How are they different? Find a problem in the text that is similar to examples 2, 3, and 4. Post the problem for your classmates to solve.

1. How does the author determine what the first equation should be?

|The first equation is the equation formed out of the first information in the problem. If the variables in the question are taken as x and y, |
|usually the first equation will be x + y = k, where k is the sum of x and y as given in the question. This step is called “Total value” step |
|and the resulting equation is called “Total Value” equation. |

2. What about the second equation?

|The second equation is the equation formed out of the second information in the problem, where the required mixture of the two components x |
|and y is formed using the data given in the problem. This step is called the “Mixture” step and the resulting equation is called “Mixture” |
|equation. |

3. How are these examples similar?

|These examples are similar in the sense that there are two unknowns in each question. They contain two sets of information that are translated|
|into two linear equations. Thus, they give us a system of linear simultaneous equations, which is then solved to find the values of both the |
|unknowns. |

4. How are they different?

|The examples are different in the type of data they possess. The problems in this category could be on two investments, two chemicals mixture |
|or on coins (pennies, nickels, dimes and quarters). How we form the two equations for a question depends upon how is the information stated in|
|the problem. |

Find a problem in the text that is similar to examples 2, 3, and 4. Post the problem for your classmates to solve.

|Here is a problem for my classmates to solve: |
|Soybean meal is 16% protein and corn meal is 9% protein. How many pounds of each should be mixed to get a 350-lb mixture that is 12% protein? |
| |
|Solution: |
|Let there be x pounds of corn meal and y pounds of soybean meal. |
|By data, x + y = 350 ( y = 350 – x |
|0.09x + 0.16y = 0.12(350) |
|0.09x + 0.16(350 – x) = 42 |
|0.09x – 0.16x = 42 – 56 |
|-0.07x = -14 |
|x = 200 pounds and y = 350 – 200 = 150 pounds |
|200 pounds of corn meal and 150 pounds of soybean meal. |