A two-step equation is an equation that takes two inverse operations to isolate the variable. These equations are important because they are a bridge between basic arithmetic and more advanced algebra. The main goal is to find the value of the variable that makes both sides of the equation true.
A balanced scale is a useful model because every legal move must keep the two sides equal.
To solve a two-step equation, undo the operations in the reverse order of how they affect the variable. Usually, you first undo addition or subtraction, then undo multiplication or division. Whatever operation you perform on one side, you must perform on the other side as well.
After solving, substitute the answer back into the original equation to check that it works.
Key Facts
- To solve ax + b = c, first subtract b from both sides, then divide by a.
- To solve ax - b = c, first add b to both sides, then divide by a.
- Inverse operations undo each other: addition and subtraction are inverses, multiplication and division are inverses.
- Balance rule: if A = B, then A + c = B + c, A - c = B - c, A c = B c, and A / c = B / c for c not equal to 0.
- Example: 3x + 5 = 20, so 3x = 15, so x = 5.
- Check by substitution: if x = 5 in 3x + 5 = 20, then 3(5) + 5 = 20, so 20 = 20.
Vocabulary
- Two-step equation
- An equation that can be solved by using two inverse operations to isolate the variable.
- Variable
- A letter or symbol that represents an unknown number.
- Inverse operation
- An operation that undoes another operation, such as subtraction undoing addition.
- Coefficient
- The number multiplied by a variable, such as 4 in 4x.
- Solution
- The value of the variable that makes the equation true.
Common Mistakes to Avoid
- Changing only one side of the equation is wrong because it destroys the balance between the two sides.
- Undoing operations in the wrong order can lead to an incorrect value because you should reverse the order of operations around the variable.
- Forgetting negative signs gives the wrong solution because signs are part of the numbers and operations in the equation.
- Not checking the answer misses errors because substitution into the original equation is the best way to confirm the solution.
Practice Questions
- 1 Solve for x: 4x + 7 = 31.
- 2 Solve for y: 18 = 3y - 6.
- 3 Explain why subtracting 5 from only one side of the equation 2x + 5 = 13 does not keep the equation balanced.