Equations & Solutions - Sixth Grade Mathematics

An algebraic equation models a relationship of perfect balance between two mathematical expressions, where solving the equation discovers the specific value that makes the statement true.

The Balanced Scale Model

The core conceptual model of an algebraic equation is a balanced two-pan balance scale. The equal sign (=) serves as the fulcrum or balance point.

The Balanced Scale Concept:
                [ Left Side Expression ]   =   [ Right Side Expression ]
                                       ^
                                       |
                                  Fulcrum (=)

If you add, subtract, multiply, or divide something on the left pan,
you MUST perform the EXACT SAME operation on the right pan
to keep the scale in perfect balance!

What is a Solution?

A solution to an equation is a specific number that, when substituted for the variable, makes the equation a true mathematical statement. Consider the equation: x + 7 = 12. If you test x = 4: 4 + 7 = 11. Since 11 does not equal 12, x = 4 is NOT a solution. If you test x = 5: 5 + 7 = 12. Since 12 = 12, the number 5 IS a solution. The set containing all solutions to an equation is called its solution set. In linear one-step equations, there is exactly one unique solution.

Types of Equations

Equations can behave in three distinct ways depending on their mathematical structure:

+-------------------------------------------------------------------------+
|                        THREE TYPES OF EQUATIONS                         |
+-------------------+-----------------------+-----------------------------+
| Type              | Definition            | Concrete Example            |
+-------------------+-----------------------+-----------------------------+
| Conditional       | True for only one     | 2x + 3 = 11                 |
| Equation          | specific value        | (True ONLY when x = 4)      |
+-------------------+-----------------------+-----------------------------+
| Identity          | True for ALL possible | 3(x + 2) = 3x + 6           |
|                   | numbers (Infinite sol)| (True for every number!)    |
+-------------------+-----------------------+-----------------------------+
| Contradiction     | NEVER true for any    | x + 5 = x + 8               |
|                   | number (No solution!) | (A number cannot equal      |
|                   |                       | itself plus three!)         |
+-------------------+-----------------------+-----------------------------+

The Principle of Isolating the Variable

The ultimate goal when solving any equation is to isolate the variable on one side of the equal sign by itself (producing x = value). To isolate the variable, you systematically strip away numbers attached to the variable by applying inverse operations.

Exercises

Exercise 1. State whether each mathematical statement is an expression or an equation: (a) 4x - 9 = 23; (b) 7y + 15; (c) (3n + 8) / 2; (d) 50 = 2(w + 10).

Exercise 2. Explain the balanced scale model of an equation, and describe what happens if you subtract 5 from only the left side of the equation x + 5 = 12.

Exercise 3. Determine by mental inspection whether x = 6 is a solution to the equation 3x - 4 = 14.

Solutions

Solution to Exercise 1. For (a) 4x - 9 = 23: contains an equal sign; Equation. For (b) 7y + 15: no equal sign; Expression. For (c) (3n + 8) / 2: no equal sign; Expression. For (d) 50 = 2(w + 10): contains an equal sign; Equation.

Solution to Exercise 2. In the balanced scale model, the left and right expressions hold equal weight, perfectly balancing across the central equal sign (=). If you subtract 5 from only the left side, the left pan becomes lighter and rises, tipping the scale out of balance. To maintain the truth of the equality, you must subtract 5 from the right side as well (12 - 5 = 7), restoring balance and revealing that x = 7.

Solution to Exercise 3. Substitute x = 6 into the left side of 3x - 4: 3(6) - 4 = 18 - 4 = 14. Because the evaluated left side (14) matches the right side (14) identically, x = 6 is confirmed to be the true solution to the equation.