Two-Step Equations
Solve equations that require undoing addition/subtraction and then multiplication/division. Work with fractional and decimal coefficients, and apply to real-world situations modeled by two-step equations.
A lesson of Algebra 1, in Solving Linear Equations, about 30 minutes.
By the end of this lesson, you will be able to solve two-step linear equations including ones with negative and fractional coefficients, check every solution, and turn a two-step word problem into an equation.
Undo It in the Reverse of the Order It Was Built
A two-step equation has had exactly two operations done to the variable.
$$3x + 5 = 20$$
Read what happened to $x$, in order. It was multiplied by $3$, and then $5$ was added. Getting back to $x$ means reversing both, last one first.
$$\text{built: } x \to 3x \to 3x + 5$$
$$\text{unbuilt: } 3x + 5 \to 3x \to x$$
That is why you undo the addition before the multiplication. It is the order of operations run backwards, and it is the opposite of what most people try first.
The balance rule has not changed. Whatever you do, do it to both sides, so the two sides stay equal at every line.
| Do this first | Then this |
|---|---|
| Undo the addition or subtraction | Undo the multiplication or division |
Example 1: The Standard Shape
Here is the whole method on one equation, worked all the way through. Solve $3x + 5 = 20$.
- Subtract 5 from both sides, writing both sides in full 3x + 5 - 5 = 20 - 5
- Simplify each side 3x = 15
- Now divide both sides by 3 and simplify x = 5
- Check it. Put 5 into the left hand side of the original equation 20
Worked Example 1: Subtraction in the Middle
Solve $2x - 7 = 11$.
Example 2: A Negative Coefficient
The method does not change. What changes is the number of places a sign can go missing, so the check earns its keep here. Solve $-4x + 3 = -13$.
- Subtract 3 from both sides, writing both sides in full -4x + 3 - 3 = -13 - 3
- Simplify each side -4x = -16
- Divide both sides by -4 and simplify x = 4
- Check it. Put 4 into the left hand side -13
Worked Example 2: Negative Again, Your Turn
Solve $-3x + 10 = 1$.
Example 3: When the Variable Is Divided
Solve $\dfrac{x}{3} - 2 = 5$. Here $x$ was divided by $3$ and then $2$ was subtracted, so you add the $2$ back first and multiply by $3$ second.
- Add 2 to both sides, writing both sides in full x/3 - 2 + 2 = 5 + 2
- Simplify each side x/3 = 7
- Multiply both sides by 3 and simplify x = 21
- Check it. Put 21 into the left hand side 5