The two methods we have used to solve systems of equations are substitution and elimination. Either method is acceptable for solving the systems of equations that we will be working with in this lesson. Method of Substitution: 1. solve one equation for one variable (it doesn t matter which equation you choose or which variable you choose). 2. substitute the solution from step 1 into the other equation. 3. solve the new equation from step 2. 4. back substitute to solve the equation from step 1. Method of Elimination: 1. multiply at least one equation by a nonzero constant so the coefficients for one variable will be opposites (same absolute value) 2. add the equations so the variable with the opposite coefficients will be eliminated. 3. take the result from step 2 and solve for the remaining variable. 4. take the solution from step 3 and back substitute to any of the equations to solve for the remaining variable. Steps for solving applications: 1. assign variables to represent the unknown quantities 2. set-up equations using the variables from step 1 3. solve using substitution or elimination; it makes no difference which method you use Just as in the previous lesson, all of these application problems should result in a system of equations with two equations and two variables: ax + by = c { dx + ey = f The equations will usually be linear, but not always.
Example 1: Set-up a system of equations and solve using any method. A salesperson purchased an automobile that was advertised as averaging 25 miles 40 miles of gasoline in the city and on the highway. A recent sales gallon gallon trip that covered 1800 miles required 51 gallons of gasoline. Assuming that the advertised mileage estimates were correct, how many miles were driven in the city and how many miles were driven on the highway? How many gallons of gasoline were used in the city? How many gallons of gasoline were used on the highway? Write an equation to represent the total number of gallons of gasoline used on the trip. How many miles were driven in the city, assuming the advertised mileage of 25 miles gallon is correct? How many miles were driven on the highway, assuming the advertised mileage of 40 miles gallon is correct? Write an equation to represent the total number of miles driven on the trip.
Write the system of equations from Example 3 and solve: Keep in mind that setting Example 3 up so that the variables represent the number gallons of gasoline used in the city and on the highway will NOT solve the problem directly since we re asked to find the number of miles driven on the highway. However if you know the number of gallons used, and you know how miles you can travel per gallon, you can still find the number miles driven. The alternative is to change miles per gallon to gallons per mile, and have the variables represent the number of miles driven.
Example 2: Set-up a system of equations and solve using any method. For a particular linear function f(x) = mx + b, f( 2) = 11 and f(3) = 9. Find m and b.
Example 3: Set-up a system of equations and solve using any method. For a particular quadratic function f(x) = ax 2 + bx + c, f( 3) = 13 and f(2) = 7. If a = 5, find b and c.
The next two problems involve objects traveling a certain distance, at a certain rate, for certain time. Anytime I encounter an application problem involving distance, rate, and time, I will set-up a table and use the formula d = r t, which represents distance equals rate times time. Example 4: Set-up a system of equations and solve using any method. Two people can canoe 18 miles in an hour and a half when going with the current. Against the same current, the same two people can only canoe 3 miles in 1 hour. Using this information, determine how fast the people can move the canoe on their own, and how fast the current is moving. With Current Against Current d = r t
Be sure to pay attention to units when working with distance, rate, and time problems; if a rate is in terms of miles per hour, then time must be in terms of hours in order to get a distance in terms of miles miles hour hours = miles Example 5: Set-up a system of equations and solve using any method. A short airplane trip between two cities took 30 minutes when traveling with the wind. The return trip took 45 minutes when traveling against the wind. If the speed of the plane with no wind is 320 mph, find the speed of the wind and the distance between the two cities (pay attention to units). With Wind Against Wind d = r t
Answers to Examples: 1. 1400 miles ; 2. b = 3, m = 4 ; 3. b = 9, c = 31 ; 4. Person s rate = 4.5 mph ; rate of current = 1.5 mph ; 5. Distance = 192 miles, wind speed = 64 mph ;