Solving One-Step Equations with Multiplication and Division | Algebra How To Help

Solving One-Step Equations
As I wrote in my first post on solving one-step equations with addition and subtraction, it does not matter which algebra textbook you are using, if you are studying algebra at any level, you need to know how to solve equations and this is the starting point, well it is the second step in the process of learning how to solve multi-step equations. The first step in learning how to solve equations was covered in my previous post on solving one-step equations with addition and subtraction.

Review of Basic Terms

When solving equation, you have to keep the overall goal in mind, to isolate the variable on one side of the equation. An equation has three parts: left, middle and right. The equal sign is 'the middle' of an equation. For example,

x = 8.

x is on the left of the equation and 8 is on the right of the equation. The equation x = 8 is an example of an equation where the variable has been isolated on the left of the equation.

If you are given an equation such as

2x = -26.

The variable is not isolated because it has a coefficient of 2. To solve this equation, divide 2 into both sides. When showing your work, you should write a fraction bar under each side of the equation and write a 2 in both denominators, thus showing each side being divided by 2. On the left, the 2's divide out and leave the variable x and on the right-26 divided by 2 is -13. Therefore, x = -13.

Solving One-Step Equations with Multiplication

Example 1

If an equation involves the division of the variable by a constant such as x/ 7 = -4, then the multiplication would be used to solve this equation because it is the opposite operation to division. To correctly show work, you use the multiplication property of equality and multiply both sides of equation 1 by 7. On the left side of the equation the 7's divide out to give x and on the right -4 times 7 gives -28.


Example 2

Similar to example 1,  example 2 involves multiplication to solve the original equation. It differs from example because the side opposite the variable is a mixed number,

x/-3 = 4 1/3.

It is still solved with the multiplication property of equality, but the math on the right causes some students problems. The solution of this equation is found by multiplying both sides by -3. As you can see, in the image, the -3s divide out on the left of the equation and give the variable x. On the left hand side of the equation gives -3 * 4 1/3. The image gives a much clearer idea of calculations. The mixed number must be changed to an improper fraction before the multiplication can take place and changes to -3 * 13/3 which gives -13. Therefore, x = -13.






Solving One-Step Equations with Division

Example 3

Recognizing when to use and using division is easier for students. I am pretty sure it has to do with the fact the equations do not always start off looking like fractions. The equation in example 3 is

2x = -26.

If you read the left of the equation with the operation, 2 times x, instead of 2x, the opposite of times is divide. So, to solve this equation, you must divide by sides by 2. On the left of the equation, the 2s divide out leaving x and on the right -26 divided by 2 is -13. Thus, x = -13.

Example 4

When solving equations, it is good to compare new problems to to previously solved problems and to compare how they are similar and different.

-0.6x = 42

Looking back at example 3, it should be obvious the two problems are similar because the variable is being multiplied by a number. The difference is that the coefficient of the x is a rational number in example 4, whereas in example 3 it was a whole number. So, solving the equation in this example will involve the division property of equality to divide each side by -0.6. On the left, the -0.6s divide out leaving the variable x and the left hand gives -70 because 42 divided by -0.6 is -70, thus x = -70.






Should I Take 1 Step or 2 Steps?

Image 1

The equation in example 5 contains a fractional coefficient. In other words, a fraction in front of the variable. There are two ways that a fractional coefficient can be model, as can be seen in Image 1, because the fractional coefficient can be written as a number times the variable in the numerator then divided by another number, these types of equations can be solved with two steps that include one step of multiplication and one step of division.

I don't know how you are, but I do not like to show more than is necessary, so I want to let you know there is way to solve these equations with only one step! Yep! Less work to do! To use one step, all you have to do is multiply both sides of the equation by the fractional coefficient's inverse. 

Example 5

In the image, I solve the equation in two different ways. On the left, I solved it with one step and on the right I solved it with two steps.

On the Left - Notice how I multiplied each side of the equation by the fractions inverse. On the left, all of the numbers divide out to 1 and leave the variable isolated. Finding the solution is a matter of basic math.

On the Right - I multiplied both sides by 3 first to give 2x = 36, then divided both sides by 2 to get x = 18. I could have just as easily divided by 2 first.

Summing it all up, solving equations requires you to be flexible in how you deal with a given equation. Though not every equation is solved exactly the same, it does require the use of the opposite operation and the properties of equality to maintain the balance of the equation.