Math 95. Section 4.1 (a): Systems of Linear Equations in Two Variables;
Solving Systems by Graphing and Substitution. And Sec. 4.3 Applications.
I. A System of Linear Equations:
A. Solution:
Ex. 1: Determine whether (­3,3) is a solution to the system:
B. Number of Possible Solutions:
1. Inconsistent:
2. Consistent:
II. Methods of Solution:
1. Graphing:
2. Substitution:
3. Elimination:
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M95, Mod 1, Sec. 4.1/4.3, pg2
II. Solving by Graphing: good for approximation, a good visual of the situation, also a good check for your symbolic solutions.
Ex. 1: Ex. 2:
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M95, Sec. 4.1/4.3, pg 3
III. Substitution:
1. Choose one of the equations and solve it for one of the variables.
2. Substitute what you get into the other equation and solve. 3. Back solve and check your solution.
Use substitution to solve the following systems of linear equations
Ex. 3: Ex. 4:
Ex. 5:
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M95, Sec. 4.1/4.3, pg 4
III. Applications: Use substitution to solve the following:
Ex. 6: (Break Even problem) Find the number of units, x, that must be sold to break even given the Cost and Revenue functions for a business to be:
Ex. 7: (A motion problem where current affects the rate) A boat travels 18 miles down a river in 1 hour. A return trip against the current takes hours. Find the average speed of the boat in still water and the rate of the current.
Ex. 8: (A Geometry problem) The length of a garden is 8 feet longer than its width. The perimeter of the garden is 54 feet. Find the area of the garden.
Ex. 9: (Another Geometry problem) A rectangular lot whose perimeter is 320 feet is fenced along three sides. An expensive fencing along the lot's length cost $16 per foot, and an inexpensive fencing along the two side widths costs $5 per foot. The total cost of the fencing along the three sides comes to $2140. What are the lot's dimensions?
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M95, Sec. 4.1 (b)con't: Solving Systems using the Elimination Method and 4.3 applications:
I. Process:
1. Get both equations in standard form. Be sure the same variables line up under each other and the equal signs are under each other.
2. Multiply or divide one or both the equations to get the same coefficient but opposite signs on one of the variables.
3. Combine the like variables and constants in a vertical manner. One of the variables should be eliminated. And, you should be left with an equation that has the other variable in it.
4. Solve for that variable.
5. Back solve and check your solution
II. Use the elimination method to solve the following:
Ex. 1: Ex. 2:
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M95, Sec. 4.1(b)/4.3, pg. 2
Ex. 3:
III. Applications: Use the elimination method to solve the following:
Ex. 4: (A motion problem with wind affecting the rate) Two airfields, Andreas and Borders, are 400 miles apart. Borders is due east of Andreas. A plane flew from Andreas to Borders in 2 hours and then returned to Andreas in hours. If the wind blew with a constant velocity from the west during the entire round trip, find the speed of the plane in still air and the speed of the wind.
Ex. 5: (A motion problem where current affects the rate) A boat travels 18 miles down a river in 1 hour. A return trip against the current takes hours. Find the average speed of the boat in still water and the rate of the current.
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M95, Sec. 4.1(b)/4.3, pg. 3
Ex. 6: (Mixture Problem): A grocer mixes two grades of coffee that sell for $6 and $8 a pound. He has a bin that holds 50 pounds of coffee. How much of each must he buy to make a mixture that sells for $7.20 a pound?
Ex. 7: (Mixture problem with investments): A sum of $1000 is invested for one year. Some of the money is invested at 6% and the remainder at 12%. How much was invested at each percent if the total interest income from the two investments is $108?
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M95, Sec. 4.1(b)/4.3, pg 4
Ex. 8: (Another mixture problem with percents) A pharmacist need 500 milliliters of a 20% Phenobarbital solution but has only 5% and 25% Phenobarbital solutions available. Find how many milliliters of each he should mix to get the desired solution.
V. What does it mean when both my variables disappear in the process?
Ex. 9:
Ex. 10:
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