A one-step equation
is as straightforward as it sounds. You will only need to perform one step in order to solve the equation.
One goal in solving an equation
is to have only variables on one side
of the equal sign and numbers on the other side
of the equal sign. The other goal is to have the number in front of the variable
equal to one.
The strategy for getting the variable
by itself with a coefficient
of 1 involves using opposite operations. For example, to move something that is added to the other side
of the equation, you should subtract. The most important thing to remember in solving a linear equation
is that whatever you do to one side
of the equation, you MUST do to the other side. So if you subtract a number from one side, you MUST subtract the same value from the other side. You will see how this works in the examples.
We’ll begin solving equations with those that only require one-step. Once you work through the examples, you may want to link to other lessons that require more steps. Let's Practice:
Remember the goal is to have the variable by itself on one side of the equation. In this problem, that means moving the 5 to the other side of the equation. Since the 5 is added to the variable, we move it to the other side of the equation by subtracting 5. However, if we subtract 5 from the left side of the equation, we MUST also subtract 5 from the right side.
It does not matter that the variable in this equation is on the right side of the equation. The position of the variable is not an issue. Remember that the goal is to have the variable on one side by itself. It does not matter which side.
To get the variable by itself, we need to add 3 to both sides.
The variable in this equation is already on one side of the equation by itself. There is no need to add or subtract anything to both sides. However, the number in front of the variable is not 1. The -3 that is in front of the variable indicates multiplication of -3 by x. The opposite operation of multiplication is division. So we will divide both sides by -3.
You should take note of the different ways of writing the answer. In the example, we divided by -3, yet wrote the answer with the negative in front of the entire fraction, not just the 3. Each of the following fractions all mean the same thing.
The variable in this equation is already on one side by itself, but it is divided by 3. To get rid of the 3 that is attached to the variable by division, we will perform the opposite operation which is multiplication. Notice that our variable can be any letter. It does not always have to be x.
Once again, the variable is on one side by itself, but is multiplied by a -3 and divided by 5. Let’s take care of each operation separately and see what happens. First we’ll get rid of the 5 by multiplying both sides by 5. Then we’ll get rid of the -3 by dividing both sides by -3.
Rather than perform two separate steps of multiplying by 5 and then dividing by -3, it is possible to combine those operations into one step. In other words, we can multiply both sides by . The value is called the reciprocal of . The reciprocal of a number has the same sign, but the numerator and denominator are reversed. So what was on bottom, is now on top. And what was on top, is now on bottom.
If we re-work Example 5 by using the reciprocal, you can see that it will save a step in the solution process.
Each of these examples has only required one step (addition, subtraction, multiplication, or division) in order to solve it. Click on these links to learn more about solving equations with two steps