What Are Two-step Equations?
Two-step equations are mathematical equations that require only two steps to solve and find the value of the variable. In order to solve two step equations, we need to undo two operations.
Two-step equation example:
| Equation | $4x\;-\;1 = 15$ |
| Step 1 | Undo the subtraction operation. 4x – 1 + 1 = 15 + 1 4x = 16 |
| Step 2 | Undo the multiplication operation. $\frac{4x}{4} = \frac{16}{4}$ x = 4 |
Recommended Games
Two-Step Equations: Definition
Two-step equations are algebraic equations that can be solved in two steps.
Two step equations may come in different forms involving combinations of addition, subtraction, multiplication, or division operation.
Examples of two-step equations:
- $2x + 1 = 5$
- $\frac{7x}{\;-\;2} = 21$
How To Solve Two-step Equations
When solving two-step algebraic equations, the goal is to isolate the variable on one side of the equation.
To isolate the variable, we have to undo the involved operations by using their inverse operations (opposite operations) and solve for the varibale.
| Operation | Inverse Operation |
|---|---|
| Addition (+) | Subtraction (-) |
| Subtraction (-) | Addition (+) |
| Multiplication ($\times$) | Division ($\div$) |
| Division ($\div$) | Multiplication ($\times$) |
In most of the two-step equations, usually we first use addition or subtraction operation to isolate the variable and in the second step, we use multiplication or division to find the value of the variable.
Example: Solve $2x + 1 = 5$
Subtract 1 from both sides.
$2x + 1\;-\;1 = 5\;-\;1$
$2x = 4$
Divide both sides by 2.
$\frac{2x}{2} = \frac{4}{2}$
$x = 2$
What Are the Rules To Solve Two-step Equations?
To solve two step equation, we need to balance both sides of the equation using following rules:
- Undo the addition by subtracting both sides with the same number.
- Undo subtraction by adding both sides with the same number.
- Undo the multiplication by dividing both sides with the same number.
- Undo the division by multiplying the same number to both sides.
Facts about Two-step Equations
- Solving two-step equations helps build the foundation for solving multi-step algebraic equations.
- The first step in solving a two-step equation is usually undoing addition or subtraction by applying the inverse operation.
- The second step generally involves undoing multiplication or division by applying the inverse operation.
Conclusion
Two-step equation has one variable and two mathematical operations in it. Using the order of operations in reverse we can reach the solution by finding the value of an unknown variable. Even though we can use different methods to solve this two step equation, solving through reverse order of operations is the easiest way.
Solved Examples on Two-step Equations
Example 1: Solve 3x + 12 = 21.
Solution:
3x + 12 = 21
Undo the addition operation by subtracting 12 from both sides of the equation.
3x + 12 – 12 = 21 – 12
3x + 0 = 9
3x = 9
Undo the multiplication operation by dividing both sides of the equation by 3 we get,
$\frac{3}{3} x = \frac{9}{3}$
x = 3
Example 2: Sam sold one-fourth of his watch collection, and then sold 5 more watches. How many watches did Sam have in the beginning if he sold a total of 10 watches?
Solution:
Suppose that Sam had x number of watches initially in his collection.
First, he sold one-fourth of his collection, which is $\frac{x}{4}$.
Next, he sold 5 more watches.
So, total number of watched he sold $= \frac{x}{4} + 5$
It is given that Sam sold a total of 10 watches.
Thus, $\frac{x}{4} + 5 = 10$
Undo the subtraction by adding the number 5 on both sides of the equation we get,
$\frac{x}{4} + 5 \;-\; 5 = 10 \;-\; 5$
$\frac{x}{4} = 5$
Undo the division operation by multiplying number 4 on both sides of the equation we get,
$\frac{4x}{4} = 5 \times 4$
$x = 20$
Hence, Ramsey had 20 watches with him in the beginning.
Example 3: Solve 2 (x + 7) = 16.
Solution:
2 (x + 7) = 16
$\frac{2}{2} (x + 7) = \frac{16}{2}$ divide by 2 on both sides
x + 7 = 8
x + 7 – 7 = 8 – 7 subtract 7 from both sides
x = 1
Another way: If we expand this equation, we get
2x + 14 = 16
2x = 16 – 14
2x = 2
x = 1
Example 4: Solve the two step equation $\frac{x \;-\; 2}{3} = 1$
Solution:
$\frac{x \;-\; 2}{3} = 1$
$\frac{3(x \;-\; 2)}{3} = 1 \times 3$ multiplying 3 on both sides of the equation we get,
x – 2 = 3 adding 2 on both sides of the equation we get,
x – 2 + 2 = 3 + 2
x = 5
Practice Problems on Two-step Equations
Two-step Equations - Definition, Steps, Facts, Examples, FAQs
Find the value of x if 2x+3=5.
2x + 3 = 5
2x + 3 - 3 = 5 - 3
2x = 2
x = 1
Solve: - x + 1 = 12
- x + 1 = 12
- x + 1 -1 = 12 - 1
- x = 11
x = -11
What are two-step equations?
Two-step equations require two steps to solve. Thus, we need to use two operations to find the value of x.
Frequently Asked Questions on Two-Step Equations
What is a multi-step equation?
A multi-step equation contains mixed operations such as addition, subtraction, multiplication, or division. It requires two or more steps to solve the equation. We can use the reverse order of operations to solve them.
What are one-step equations?
One-step equations are algebraic equations that require only one step to solve.
How do you check if the solution to a two-step equation is correct?
To check the solution, substitute the found value back into the original equation and verify if both sides are equal.
