8.1 Solve Equations Using the Subtraction and Addition Properties of Equality

Solving Linear Equations

Linear equations let you find an unknown value by keeping both sides of an equation balanced. Mastering the addition and subtraction properties of equality gives you the foundation for every equation-solving technique you'll learn from here on out.

Properties of Equality in Linear Equations

An equation is like a balanced scale. Whatever you do to one side, you must do to the other to keep it balanced. That's the core idea behind these two properties.

• Subtraction Property of Equality: If $a = b$, then $a - c = b - c$. You can subtract the same number from both sides without breaking the equality.
• Addition Property of Equality: If $a = b$, then $a + c = b + c$. You can add the same number to both sides and the equation stays true.

The goal is to isolate the variable (get it alone on one side). Here's how you decide which property to use:

• If a number is added to the variable, subtract it from both sides.
  • $x + 3 = 7$ → subtract 3 from both sides → $x = 4$
• If a number is subtracted from the variable, add it to both sides.
  • $x - 5 = 2$ → add 5 to both sides → $x = 7$

Always substitute your answer back into the original equation to verify. For example, plugging $x = 4$ into $x + 3 = 7$ gives $4 + 3 = 7$, which is true, so the solution checks out.

Simplification of Complex Equations

Before you can isolate the variable, you often need to simplify each side of the equation first.

Combine like terms on each side. Like terms share the same variable raised to the same exponent. For instance, $3x$ and $5x$ are like terms, but $3x$ and $5x^2$ are not.

Follow the order of operations when simplifying:

1. Parentheses
2. Exponents
3. Multiplication and division (left to right)
4. Addition and subtraction (left to right)

Once both sides are simplified, use the addition or subtraction properties to isolate the variable.

Example: Solve $4x + 2 - x = 11$

1. Combine like terms on the left: $3x + 2 = 11$
2. Subtract 2 from both sides: $3x = 9$
3. (You'd divide both sides by 3 here, but that uses a property covered in a later section.)

Translating Word Problems into Equations

Turning a word problem into an equation is a skill that takes practice. Here's a reliable process:

1. Read the problem carefully and identify what's unknown. Assign a variable (like $x$) to represent it.

2. Translate key phrases into math operations:

   • "sum" or "more than" → addition
   • "difference" or "less than" → subtraction
   • "product" or "times" → multiplication
   • "quotient" or "divided by" → division

3. Write the equation using the relationships described in the problem.

4. Solve using the properties of equality.

5. Interpret your answer in the context of the problem. Write it as a sentence, not just a number.

Example: "A number increased by 8 is 15. Find the number."

• Let $x$ = the unknown number.
• "Increased by 8" means add 8: $x + 8 = 15$
• Subtract 8 from both sides: $x = 7$
• Answer: The number is 7.

Real-World Applications of Linear Equations

Real-world problems follow the same translation process, but the context matters more. After solving, always ask yourself: does this answer make sense?

For example, if a problem asks about the length of a fence and you get a negative number, something went wrong. State your final answer in a complete sentence that connects back to the problem, like "The length of the rectangle is 12 cm."

Checking Solutions and Problem-Solving Strategies

Checking Solutions by Substitution

Never skip this step. Substituting your answer back into the original equation is the fastest way to catch mistakes.

1. Replace the variable in the original equation with your solution.
2. Simplify both sides separately.
3. If the left side equals the right side, your solution is correct.

Example: Check whether $x = 4$ solves $2x + 3 = 11$.

• Left side: $2(4) + 3 = 8 + 3 = 11$
• Right side: $11$
• $11 = 11$ ✓ The solution is correct.

The solution set is the collection of all values that make the equation true. For the linear equations in this unit, there's typically one solution.

Problem-Solving Strategies for Challenging Equations

When a problem feels overwhelming, break it down:

• Identify what you know and what you're solving for.
• Simplify each side of the equation before trying to isolate the variable.
• Use the properties of equality one step at a time.
• Check your solution and make sure it makes sense in context.
• If you're stuck, try re-reading the problem or working through a similar example first.

Expressions vs. Equations

These two terms come up constantly, so know the difference:

• An algebraic expression combines variables, numbers, and operations but has no equals sign. Example: $3x + 5$
• An equation states that two expressions are equal, connected by an equals sign. Example: $3x + 5 = 14$

Solving an equation means finding the value(s) of the variable that make both sides equal. You can simplify an expression, but you solve an equation.