5.4 Solve Equations with Decimals

Solving Equations with Decimals

Solving equations with decimals works exactly like solving equations with whole numbers. The only difference is that your arithmetic involves decimal values. If you can solve $x + 3 = 10$, you can solve $x + 3.2 = 10.7$.

This section covers how to verify solutions, isolate variables in decimal equations, and translate word problems into equations you can solve.

Key Components of Algebraic Equations

Before diving in, make sure you're comfortable with these terms:

• Variable: A letter (like $x$, $y$, or $z$) that represents an unknown value you're solving for.
• Coefficient: The number multiplied by a variable. In $2.5x$, the coefficient is $2.5$.
• Inverse operations: Operations that "undo" each other. Addition undoes subtraction, and multiplication undoes division. These are your main tools for isolating a variable.

Verifying a Solution

Before you learn to solve equations, it helps to know how to check whether a given value actually works. Verification is straightforward:

1. Take the value you think is the solution and plug it in for the variable.
2. Simplify each side of the equation separately.
3. If both sides are equal, the value is a valid solution. If not, it isn't.

Example: Does $x = 2.9$ satisfy $2x + 1.5 = 7.3$?

1. Substitute: $2(2.9) + 1.5$
2. Simplify: $5.8 + 1.5 = 7.3$
3. The left side equals $7.3$, which matches the right side. So yes, $x = 2.9$ is the solution.

Get in the habit of checking your answers this way. It only takes a few seconds and catches arithmetic mistakes before they cost you points.

Solving Decimal Equations

The goal is always the same: get the variable alone on one side. You do this by using inverse operations on both sides of the equation.

Example: Solve $0.5x - 1.2 = 3.8$

1. The $-1.2$ is keeping $x$ from being isolated. Add $1.2$ to both sides: $0.5x = 5.0$
2. The variable $x$ is being multiplied by $0.5$. Divide both sides by $0.5$ (or equivalently, multiply by $2$): $x = 10$
3. Check: $0.5(10) - 1.2 = 5.0 - 1.2 = 3.8$ ✓

Example: Solve $x + 4.75 = 12.3$

1. Subtract $4.75$ from both sides: $x = 7.55$
2. Check: $7.55 + 4.75 = 12.3$ ✓

A few tips to keep your work clean:

• Line up your decimal points when adding or subtracting vertically.
• When dividing by a decimal like $0.25$, you can think of it as multiplying by its reciprocal ($4$) if that feels easier.
• Always bring your solution back to the original equation to verify, not a simplified version partway through your work.

Word Problems with Decimals

Word problems just add one extra step at the beginning: you need to translate the English into an equation. After that, you solve it the same way.

1. Identify the unknown and assign it a variable.

2. Translate the words into math. Watch for keywords:

   • "is" or "are" → $=$
   • "more than" → addition
   • "less than" → subtraction
   • "times" or "each" → multiplication
   • "divided by" or "split among" → division

3. Solve the equation using inverse operations.

4. Check that your answer makes sense in context. (You can't buy 3.7 shirts, for instance.)

Example: A store sells shirts for $\$15.99$ each. If the total cost is $\$79.95$, how many shirts were purchased?

1. Let $x$ = the number of shirts.
2. Write the equation: $15.99x = 79.95$
3. Divide both sides by $15.99$: $x = 5$
4. Check: $15.99 \times 5 = 79.95$ ✓ — and 5 is a whole number, which makes sense for a count of shirts.

Example: After spending $\$8.50$ on lunch, you have $\$14.25$ left. How much did you start with?

1. Let $x$ = your starting amount.

2. Equation: $x - 8.50 = 14.25$

3. Add $8.50$ to both sides: $x = 22.75$

4. Check: $22.75 - 8.50 = 14.25$ ✓