# Systems of Linear Equations

## Solve Applications with Systems of Equations


### Solve Direct Translation Applications

Systems of linear equations are very useful for solving applications. Some people find setting up word problems with two variables easier than setting them up with just one variable. To solve an application, we’ll first translate the words into a system of linear equations. Then we will decide the most convenient method to use, and then solve the system.
We solved number problems with one variable earlier. Let’s see how differently it works using two variables.
As you solve each application, remember to analyze which method of solving the system of equations would be most convenient.

### Solve Geometry Applications

We will now solve geometry applications using systems of linear equations. We will need to add complementary angles and supplementary angles to our list some properties of angles.
The measures of two complementary angles add to 90 degrees. The measures of two supplementary angles add to 180 degrees.
If two angles are complementary, we say that one angle is the complement of the other.
If two angles are supplementary, we say that one angle is the supplement of the other.
In the next example, we remember that the measures of supplementary angles add to 180.
Recall that the angles of a triangle add up to 180 degrees. A right triangle has one angle that is 90 degrees. What does that tell us about the other two angles? In the next example we will be finding the measures of the other two angles.
Often it is helpful when solving geometry applications to draw a picture to visualize the situation.

### Solve uniform motion applications

We used a table to organize the information in uniform motion problems when we introduced them earlier. We’ll continue using the table here. The basic equation was  where D is the distance traveled, r is the rate, and t is the time.
Our first example of a uniform motion application will be for a situation similar to some we have already seen, but now we can use two variables and two equations.
Many real-world applications of uniform motion arise because of the effects of currents—of water or air—on the actual speed of a vehicle. Cross-country airplane flights in the United States generally take longer going west than going east because of the prevailing wind currents.
Let’s take a look at a boat travelling on a river. Depending on which way the boat is going, the current of the water is either slowing it down or speeding it up.
The images below show how a river current affects the speed at which a boat is actually travelling. We’ll call the speed of the boat in still water b and the speed of the river current c.
The boat is going downstream, in the same direction as the river current. The current helps push the boat, so the boat’s actual speed is faster than its speed in still water. The actual speed at which the boat is moving is 
Now, the boat is going upstream, opposite to the river current. The current is going against the boat, so the boat’s actual speed is slower than its speed in still water. The actual speed of the boat is 
We’ll put some numbers to this situation in the next example.
Wind currents affect airplane speeds in the same way as water currents affect boat speeds. We’ll see this in the next example. A wind current in the same direction as the plane is flying is called a tailwind. A wind current blowing against the direction of the plane is called a headwind.

### Key Concepts

1. How To Solve Applications with Systems of Equations



### Practice Makes Perfect

Direct Translation Applications
In the following exercises, translate to a system of equations and solve.
Solve Geometry Applications
In the following exercises, translate to a system of equations and solve.
Solve Uniform Motion Applications
In the following exercises, translate to a system of equations and solve.

### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all objectives?