2.3 Solving Equations Using the Subtraction and Addition Properties of Equality

Solving equations is all about balance. The subtraction and addition properties of equality let you add or subtract the same amount from both sides of an equation, keeping things equal while you isolate the variable.

These properties show up constantly in algebra, and they're also how you solve real-world problems. Once you can translate a word problem into an equation and solve it, you can figure out things like unknown prices, distances, or quantities.

Solving Equations Using the Subtraction and Addition Properties of Equality

Equation Verification Process

Before solving equations, you need to know how to check whether a given value actually makes an equation true. This is called verifying a solution.

1. Substitute the given number for the variable in the equation. For example, if you're checking whether $x = 3$ is a solution to $x + 2 = 5$, replace every $x$ with 3.
2. Simplify the left side by performing the operations. Here, $3 + 2 = 5$.
3. Simplify the right side the same way. The right side is already 5.
4. Compare both sides.
   • If they're equal, the number is a solution. Since $5 = 5$, $x = 3$ satisfies $x + 2 = 5$.
   • If they're not equal, it's not a solution. For instance, $3 + 2 = 5 \neq 6$, so $x = 3$ does not satisfy $x + 2 = 6$.

You'll use this same checking process after solving any equation to make sure your answer is correct.

Subtraction Property in Equations

The Subtraction Property of Equality says: if you subtract the same number from both sides of an equation, the equation stays true.

Use this when a number is being added to the variable and you need to get rid of it.

How to apply it:

1. Look at what's being added to the variable. In $x + 5 = 9$, the number 5 is added to $x$.

2. Subtract that number from both sides: $x + 5 - 5 = 9 - 5$.

3. Simplify: $x = 4$.

4. Check by plugging back in: $4 + 5 = 9$. ✓

The key idea is that adding 5 and then subtracting 5 cancel each other out, leaving $x$ by itself.

Addition Property in Equations

The Addition Property of Equality says: if you add the same number to both sides of an equation, the equation stays true.

Use this when a number is being subtracted from the variable and you need to undo it.

How to apply it:

1. Look at what's being subtracted from the variable. In $x - 3 = 5$, the number 3 is subtracted from $x$.

2. Add that number to both sides: $x - 3 + 3 = 5 + 3$.

3. Simplify: $x = 8$.

4. Check by plugging back in: $8 - 3 = 5$. ✓

Subtracting 3 and then adding 3 cancel out, isolating $x$.

Word Problems to Algebraic Equations

Translating a word problem into an equation takes practice, but there's a reliable process:

1. Identify the unknown and assign it a variable. For example, "a number" becomes $x$.

2. Look for keywords that tell you which operation to use:

   • sum, more than, increased by → addition
   • difference, less than, decreased by → subtraction
   • product, times → multiplication
   • quotient, divided by → division

3. Build the equation by connecting two expressions with an equals sign. If the problem says "the sum of a number and 5 is 12," that translates to $x + 5 = 12$.

4. Solve using the addition or subtraction property. Here, subtract 5 from both sides: $x = 7$.

Real-World Equation Applications

Word problems in context follow the same steps, with one extra layer: you need to interpret your answer.

Example: You bought a book and a notebook. The notebook cost $4, and the total was $13. How much was the book?

1. Assign a variable: let $x$ = the price of the book.
2. Write the equation: $x + 4 = 13$.
3. Solve: subtract 4 from both sides → $x = 9$.
4. Interpret: the book cost $9.
5. Check: $9 + 4 = 13$. ✓

Always re-read the original problem after solving to make sure your answer actually makes sense in context.

Addition vs. Subtraction in Equations

Choosing which property to use comes down to one question: what operation do you need to undo?

• If a number is added to the variable (like $x + 7 = 10$), subtract it from both sides.
• If a number is subtracted from the variable (like $x - 4 = 2$), add it to both sides.

You're always doing the opposite (inverse) operation to cancel out the extra term and leave the variable alone. Think of it as "undoing" whatever was done to $x$.

Mathematical Properties and Equation Solving Strategies

A few properties from arithmetic carry over into algebra and help you work with equations:

• The commutative property means order doesn't matter for addition or multiplication: $a + b = b + a$.
• The associative property means grouping doesn't matter: $(a + b) + c = a + (b + c)$.
• Inverse operations are the core strategy here. Addition and subtraction are inverses of each other, so you use one to undo the other.

As equations get more complex, you'll also combine like terms and use multiple steps. For now, focus on getting comfortable with the one-step process of adding or subtracting to isolate the variable. That foundation supports everything that comes next.