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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:
- solve one equation for one variable (it doesnโt matter which equation
you choose or which variable you choose).
- substitute the solution from step 1 into the other equation.
- solve the new equation from step 2.
- back substitute to solve the equation from step 1.
Method of Elimination:
- multiply at least one equation by a nonzero constant so the
coefficients for one variable will be opposites (same absolute value)
- add the equations so the variable with the opposite coefficients will
be eliminated.
- take the result from step 2 and solve for the remaining variable.
- take the solution from step 3 and back substitute to any of the
equations to solve for the remaining variable.
Steps for solving applications:
- assign variables to represent the unknown quantities
- set-up equations using the variables from step 1
- 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:
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
gallon
of gasoline in the city and
40 miles
gallon
on the highway. A recent sales
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?
25
miles
gallon
โ ๐ฅ gallons =
How many miles were driven on the highway, assuming the advertised
mileage of
40 miles
gallon
is correct?
40
miles
gallon
โ ๐ฆ gallons =
Write an equation to represent the total number of miles driven on the trip.
Example 2 : Set-up a system of equations and solve using any method.
For a particular linear function ๐
= 11 and
= โ 9. Find the values of ๐ and ๐.
Example 3 : Set-up a system of equations and solve using any method.
For a particular quadratic function ๐
2
and ๐
= 7. Find the values of ๐ and ๐, given that ๐ = 5.
2
2
2
2
At this point I have two equations, but both equations have three
unknowns
๐, ๐, and ๐
. Since weโre told in the direction that ๐ = 5 , I
can replace ๐ with 5 in both equations, and this will leave me with two
equations with only two unknowns.
Since ๐ฅ represents how fast the people can move the boat, that means the
people can row the boat 7. 5 miles per hour. To find the value of ๐ฆ (the
speed of the current), Iโll back substitute to one of the prior equations by
replacing ๐ฅ with 7. 5.
Since ๐ฆ represents the speed of the current, that means the current is
moving at 4. 5 miles per hour.
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
โ hours = miles
This will be important on Example 5, because we will need to convert our
time units from minutes to hours.
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 (๐) and the distance (๐
) between the two cities (pay attention to
units).
miles =
miles
hour
โ hours
With Wind
Against Wind
