Applications of Linear Equations

Objective: Solve number and geometry problems by creating and solving a linear equation.

Word problems can be tricky. Often it takes a bit of practice to convert the English sentence into a mathematical sentence. This is what we will focus on here with some basic number problems, geometry problems, and parts problems.

A few important phrases are described below that can give us clues for how to set up a problem.

• A number (or unknown, a value, etc) often becomes our variable
• Is (or other forms of is: was, will be, are, etc) often represents equals (=) $x$ is 5 becomes $x =5$
• More than often represents addition and is usually built backwards, writing the second part plus the first
  Three more than a number becomes $x + 3$
• Less than often represents subtraction and is usually built backwards as well, writing the second part minus the first
  Four less than a number becomes $x − 4$

Using these key phrases we can take a number problem and set up and equation and solve.

Source: Tyler Wallace, http://www.wallace.ccfaculty.org/book/Beginning_and_Intermediate_Algebra.pdf
This work is licensed under a Creative Commons Attribution 3.0 License.

If $28$ less than five times a certain number is $232$. What is the number?

This same idea can be extended to a more involved problem as shown in the next example.

Fifteen more than three times a number is the same as ten less than six times the number. What is the number?

Another type of number problem involves consecutive numbers. Consecutive numbers are numbers that come one after the other, such as 3, 4, 5. If we are looking for several consecutive numbers it is important to first identify what they look like with variables before we set up the equation. This is shown in the following example.

The sum of three consecutive integers is $93$. What are the integers?

Sometimes we will work consecutive even or odd integers, rather than just consecutive integers. When we had consecutive integers, we only had to add $1$ to get to the next number so we had $x$, $x + 1$, and $x + 2$ for our first, second, and third number respectively. With even or odd numbers they are spaced apart by two. So if we want three consecutive even numbers, if the first is $x$, the next number would be $x + 2$, then finally add two more to get the third, $x + 4$. The same is true for consecutive odd numbers, if the first is $x$, the next will be $x + 2$, and the third would be $x + 4$. It is important to note that we are still adding $2$ and $4$ even when the numbers are odd. This is because the phrase "odd" is referring to our $x$, not to what is added to the numbers. Consider the next two examples.

The sum of three consecutive even integers is $246$. What are the numbers?

Find three consecutive odd integers so that the sum of twice the first, the second and three times the third is $152$.

When we started with our first, second, and third numbers for both even and odd we had $x$, $x + 2$, and $x + 4$. The numbers added do not change with odd or even, it is our answer for $x$ that will be odd or even.

Another example of translating English sentences to mathematical sentences comes from geometry. A well known property of triangles is that all three angles will always add to $180$. For example, the first angle may be $50$ degrees, the second $30$ degrees, and the third $100$ degrees. If you add these together, $50 + 30 + 100 = 180$. We can use this property to find angles of triangles.

World View Note: German mathematician Bernhart Thibaut in 1809 tried to prove that the angles of a triangle add to 180 without using Euclid’s parallel postulate (a point of much debate in math history). He created a proof, but it was later shown to have an error in the proof.

The second angle of a triangle is double the first. The third angle is $40$ less than the first. Find the three angles.

Another geometry problem involves perimeter or the distance around an object. For example, consider a rectangle has a length of 8 and a width of 3. There are two lengths and two widths in a rectangle (opposite sides) so we add $8 + 8 + 3 + 3 = 22$. As there are two lengths and two widths in a rectangle an alternative to find the perimeter of a rectangle is to use the formula $P = 2L + 2W$. So for the rectangle of length 8 and width 3 the formula would give, $P = 2(8) + 2(3) = 16 + 6 = 22$. With problems that we will consider here the formula $P = 2L + 2W$ will be used.

The perimeter of a rectangle is $44$. The length is $5$ less than double the width. Find the dimensions.

We have seen that it is important to start by clearly labeling the variables in a short list before we begin to solve the problem. This is important in all word problems involving variables, not just consecutive numbers or geometry problems. This is shown in the following example.

A sofa and a love seat together costs $$444$. The sofa costs double the love seat. How much do they each cost?

Be careful on problems such as these. Many students see the phrase "double" and believe that means we only have to divide the $444$ by $2$ and get $$222$ for one or both of the prices. As you can see this will not work. By clearly labeling the variables in the original list we know exactly how to set up and solve these problems.