Equations and Solutions: What It Means to Solve for a Variable

The equals sign means “has the same value as”

An equation is a statement that two expressions are equal. The symbol = does not mean “do something now” or “write the answer.” It means the left side and the right side represent the same value.

Think of an equation like a balance scale: if both sides have the same total weight, the scale is balanced. If one side is heavier, the statement is false.

Start with numeric examples (no variables yet)

Decide whether each equation is true by evaluating both sides.

• 7 = 7 is true because both sides are 7.
• 3 + 4 = 8 is false because the left side is 7 and the right side is 8.
• 12 - 5 = 2 + 5 is true because 12 - 5 = 7 and 2 + 5 = 7.

A helpful habit is to read an equation as: “left side has the same value as right side.”

What it means to solve an equation

When an equation contains a variable, a solution is a value of the variable that makes the equation true.

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Example: In the equation x + 3 = 10, the solution is x = 7 because replacing x with 7 makes the statement true.

Checking a solution by substitution

Substitution means replacing the variable with a number and then evaluating both sides.

Steps to check whether a number is a solution:

• Step 1: Substitute the candidate value for the variable.
• Step 2: Evaluate the left side.
• Step 3: Evaluate the right side.
• Step 4: Compare. If both sides match, it is a solution. If not, it is not a solution.

Guided examples: testing candidate solutions

Example 1: One-step equation

Equation: x + 5 = 14

Test the candidate x = 9:

• Substitute: 9 + 5 = 14
• Left side: 9 + 5 = 14
• Right side: 14
• Both sides are 14, so x = 9 is a solution.

Test the candidate x = 8:

• Substitute: 8 + 5 = 14
• Left side: 13
• Right side: 14
• 13 ≠ 14, so x = 8 is not a solution.

Example 2: Variable on both sides

Equation: 2x + 1 = x + 7

Test x = 6:

• Left side: 2(6) + 1 = 12 + 1 = 13
• Right side: 6 + 7 = 13
• Both sides match, so x = 6 is a solution.

Test x = 5:

• Left side: 2(5) + 1 = 10 + 1 = 11
• Right side: 5 + 7 = 12
• 11 ≠ 12, so x = 5 is not a solution.

Example 3: Parentheses and order of operations

Equation: 3(x - 2) = 12

Test x = 6:

• Left side: 3(6 - 2) = 3(4) = 12
• Right side: 12
• Matches, so x = 6 is a solution.

Test x = 5:

• Left side: 3(5 - 2) = 3(3) = 9
• Right side: 12
• 9 ≠ 12, so x = 5 is not a solution.

Equality means both sides matter (common pitfall)

A common mistake is to treat = like a signal that “the answer comes next,” focusing on only one side. In an equation, both sides are expressions that must stay equal.

Spot the issue

Consider the chain: 8 + 4 = 12 + 3 = 15

This is incorrect because it claims 12 + 3 has the same value as 8 + 4. But 8 + 4 = 12 and 12 + 3 = 15, and 12 ≠ 15.

Better ways to write the thinking:

• 8 + 4 = 12, then 12 + 3 = 15
• Or: 8 + 4 + 3 = 15

Use a “both sides” check

Whenever you see an equation, ask: “What is the value of the left side? What is the value of the right side?” If they match, the equation is true for that value of the variable.

Practice: write an equation, then check a proposed solution

For each item: (1) write an equation, (2) substitute the proposed value, (3) decide whether it is a solution by comparing both sides.

Practice Set A

1. Sentence: “A number increased by 9 is 20.” Proposed solution: x = 11

   • Equation: x + 9 = 20
   • Check: 11 + 9 = 20 → 20 = 20 (true)
   • So x = 11 is a solution.

2. Sentence: “Five times a number is 35.” Proposed solution: x = 6

   • Equation: 5x = 35
   • Check: 5(6) = 35 → 30 = 35 (false)
   • So x = 6 is not a solution.

3. Sentence: “The difference between a number and 4 is 9.” Proposed solution: x = 13

   • Equation: x - 4 = 9
   • Check: 13 - 4 = 9 → 9 = 9 (true)
   • So x = 13 is a solution.

4. Sentence: “Twice a number plus 1 equals 15.” Proposed solution: x = 8

   • Equation: 2x + 1 = 15
   • Check: 2(8) + 1 = 15 → 17 = 15 (false)
   • So x = 8 is not a solution.

Practice Set B (focus on equality, not just computing)

1. Sentence: “Three more than a number equals the number plus 3.” Proposed solution: x = 100

   • Equation: x + 3 = x + 3
   • Check: substitute 100 → 100 + 3 = 100 + 3 → 103 = 103 (true)
   • So x = 100 is a solution (in fact, every number works because both sides are identical).

2. Sentence: “A number plus 6 equals 20.” Proposed solution: x = 12

   • Equation: x + 6 = 20
   • Check: 12 + 6 = 20 → 18 = 20 (false)
   • So x = 12 is not a solution.

3. Sentence: “Four times a number minus 2 equals 18.” Proposed solution: x = 5

   • Equation: 4x - 2 = 18
   • Check: 4(5) - 2 = 18 → 20 - 2 = 18 → 18 = 18 (true)
   • So x = 5 is a solution.

4. Sentence: “A number divided by 3 equals 4.” Proposed solution: x = 15

   • Equation: x/3 = 4
   • Check: 15/3 = 4 → 5 = 4 (false)
   • So x = 15 is not a solution.

Quick self-check table: is the equation true?