A system of equations is a collection of two or more equations with the same set of
unknowns. In solving a system of equations, we try to find values for each of the
unknowns that will satisfy every equation in the system.
The equations in the system can be linear or non-linear. This tutorial reviews systems of
linear equations.
A problem can be expressed in narrative form or the problem can be expressed in
algebraic form.
Let's start with an example stated in narrative form. We'll convert it to an equivalent
equation in algebraic form, and then we will solve it.
Problem 2.1b:
An airplane flying into a head wind travels the 1800-mile flying distance between two
cities in 3 hours and 36 minutes. On the return flight, the same distance is traveled in 3
hours. Find the ground speed of the plane and the speed of the wind, assuming that both
remain constant. [Ground speed is the speed of the plane if there were no wind.]
Answer: The ground speed of the plane is 550 miles per hour and the wind speed is
50 miles per hour.
Solutions: The relationship between distance, speed, and time is distance equals speed
times time. Let the symbol d represent distance, the symbol r represent speed (or rate),
and the symbol t represent the time. The relationship between the three can then be
expressed algebraically.
The net speed has two components: ground speed of the plane and wind speed (velocity),
where ground speed represents the speed of the plane without wind. When the plane is
flying into a head wind (wind pushing against plane), the net speed of the plane is the
ground speed of the plane minus the wind speed. When the plane is flying with a tail
wind (wind pushing plane), the net speed of the plane is the ground speed of the plane
plus the wind speed. For this problem, we will assume that the plane and the wind are in
line. When you get into the wind pushing against the plane at, say a thirty-degree angle,
you solve it with vectors.
When the plane's speed is helped by the wind, the net speed increases and the time to
reach destination decreases. When the plane's is hindered by the wind, the net speed
decreases and the time to reach destination increases. Therefore, we know that the plane
had a tail wind when the time is 3 hours, and the plane had a head wind when the time is
3 hours and 36 minutes.
Let's rewrite the problem.
With head wind: distance = (plane ground speed - wind speed)
1,800= (plane speed - wind speed)
3 hours and 36 minutes
With tail wind: distance = (plane speed + wind speed)
1,800= (plane speed + wind speed)
time or
time or
3 hours
Let the symbol x represent the phrase plane ground speed and let the symbol y represent
the phrase wind speed, convert 3 hours and 36 minutes to 3.6 hours, and rewrite the two
equations in algebraic form.
x - y = 3.6
x + y = 3.0
We have converted a narrative statement of the problem to an equivalent algebraic
statement of the problem. Let's solve this system of equations.
A system of linear equations can be solved four different ways:
Substitution
Elimination
Matrices
Graphing
The Method of Substitution:
The method of substitution involves several steps:
Step 1:
Solve for x in equation (1).
Step 2:
Substitute this value for x in equation (2). This will change equation (2) to an equation
with just one variable, y.
Step 3:
Solve for y in the translated equation (2).
Step 4:
Substitute this value of y in equation (1) and solve for x.
The ground speed of the plane was 500 miles per hour and the wind speed was 50 miles
per hour.
Step 5:
Check your answers by substituting the values of x and y in each of the original
equations. If, after the substitution, the left side of the equation equals the right side of the
equation, you know that your answers are correct.