Learn: a notebook purchase
Three identical notebooks and a $2 folder cost $14. Let x be the price of one notebook, in dollars. The equation is 3x + 2 = 14: the left side describes the items and the right side gives the total cost.
Words and symbols
An equation states that two expressions are equal. A solution is a value that makes that statement true. In a one-variable linear equation, simplifying gives a form ax + b = c, with constant numbers a, b and c. When a is nonzero, the equation has one solution. Special cases can have no solution or all real numbers as solutions.
In this article x is the only variable. A two-variable equation such as y = 3x + 2 describes a line in an x–y graph; it is a different task from finding the one notebook price.
Worked example: undo the extra cost, then the multiplication
- Start with 3x + 2 = 14.
- Subtract 2 from both sides: 3x + 2 − 2 = 14 − 2, so 3x = 12.
- Divide both sides by the same nonzero number, 3: x = 4.
- Check in the original equation: 3(4) + 2 = 12 + 2 = 14.
- Interpret: each notebook costs $4.
Doing the same addition or subtraction on each side preserves equality. Dividing both sides by a nonzero number also preserves it. “Move it across and change the sign” is a shortcut for these operations, not the reason they work.
Explore: which value makes the statement true?
Choose a possible notebook price. Compare the calculated cost with $14. The equation is satisfied only when the two amounts are equal.
x = $4: left side $14, right side $14. Equal: x = 4 is the solution.
Worked example: variables on both sides
Solve 2x − 7 = 5x + 1.
- Subtract 2x from both sides: −7 = 3x + 1.
- Subtract 1 from both sides: −8 = 3x.
- Divide both sides by 3: x = −8/3.
- Check: the left side is −16/3 − 21/3 = −37/3. The right side is −40/3 + 3/3 = −37/3. They agree.
Combining like terms within an expression is different from removing a term from one side of an equation. If a term is on the other side of the equals sign, explain the operation applied to both sides.
When the variable disappears
For 2(x + 1) = 2x + 2, both sides simplify to 2x + 2. Subtracting that expression leaves 0 = 0, so every real number works. For 2(x + 1) = 2x + 5, subtraction instead leaves 2 = 5: no value works. Neither situation permits division by zero.
Practice: solve and check
1. Solve 4x − 5 = 11.
Add 5 to both sides: 4x = 16. Divide by 4: x = 4. Check: 16 − 5 = 11.
2. Solve 3(x − 2) = 9.
Divide both sides by 3: x − 2 = 3. Add 2: x = 5. Check: 3(5 − 2) = 9. Distributing first is also valid.
3. Solve 5x + 2 = 5x − 1.
Subtract 5x: 2 = −1, a false statement. There is no solution.
4. Solve 2x + 6 = 2(x + 3).
The right side expands to 2x + 6. The equation is true for all real x.
Review: equality is the rule
Write what x represents, simplify each side, show an operation on both sides, and substitute your result into the original equation. A correct answer with an unexplained sign change is a reason to revisit the reasoning. Review simplifying expressions or explore Algebra 1.