One-Step Equations
Solve equations requiring a single inverse operation. Emphasize the balance model: whatever is done to one side must be done to the other, and always verify the solution by substitution.
A lesson of Algebra 1, in Solving Linear Equations, about 30 minutes.
Solving One-Step Equations
By the end of this lesson, you will be able to solve any equation that needs exactly one inverse operation, handle negative and fractional coefficients, and check every solution by substituting it back.
Both Sides Weigh the Same, So Do Everything Twice
An equation is a sentence claiming two quantities are equal.
$$x + 7 = 12$$
Picture a balance scale with $x + 7$ on one pan and $12$ on the other. It is level, because that is what the equals sign says.
Now take $7$ off the left pan. The scale tips, and the sentence stops being true. Take $7$ off both pans and it stays level, and the sentence stays true. That is the whole rule, and every equation you ever solve obeys it.
$$\text{whatever you do to one side, do to the other}$$
Your goal is to get the variable alone on one side. To strip away what is attached to it, you apply the operation that undoes it.
| Operation | What undoes it |
|---|---|
| Adding $n$ | Subtracting $n$ |
| Subtracting $n$ | Adding $n$ |
| Multiplying by $n$ | Dividing by $n$ |
| Dividing by $n$ | Multiplying by $n$ |
A one-step equation is one where exactly one operation has been done to the variable. Undo that one thing and you are finished.
The Four Shapes
Something was added
$$x + 7 = 12 \quad \text{so subtract } 7 \text{ from both sides} \quad x = 5$$
Something was subtracted
$$n - 4 = 9 \quad \text{so add } 4 \text{ to both sides} \quad n = 13$$
The variable was multiplied
$$8x = 56 \quad \text{so divide both sides by } 8 \quad x = 7$$
The variable was divided
$$\frac{y}{6} = 3 \quad \text{so multiply both sides by } 6 \quad y = 18$$
Four shapes, one idea. Look at what was done to the variable, then do the opposite to both sides.
Example 1: Undoing an Addition
Here is the whole method on one equation, worked all the way through. Solve $x + 7 = 12$. Seven has been added to $x$, so subtracting seven is what takes it off. Doing it to both sides is what keeps the equation true.
- Subtract 7 from both sides, writing both sides in full x + 7 - 7 = 12 - 7
- Simplify each side x = 5
- Check it. Put 5 back into the left hand side and work it out 12
Worked Example 1: Undoing a Subtraction
Solve $n - 4 = 9$.