Algebra 1 Practice - Solving an Equation with Variables on Both Sides (Example 5)

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Solving an equation with variables on both sides involves isolating the variable on one side of the equation. Here’s a step-by-step guide:
General Steps:
1. **Simplify Each Side**: If necessary, simplify both sides of the equation by combining like terms and using the distributive property.
2. **Move Variables to One Side**: Use addition or subtraction to move all variable terms to one side of the equation.
3. **Move Constants to the Other Side**: Use addition or subtraction to move all constant terms to the opposite side of the equation.
4. **Solve for the Variable**: Simplify the resulting one-step or two-step equation to solve for the variable.
5. **Check Your Solution**: Substitute your solution back into the original equation to verify it makes the equation true.
Example 1: Simple Equation with Variables on Both Sides
**Equation**: \(3x + 5 = 2x + 9\)
1. *Subtract \(2x\) from both sides* to move the variable terms to one side:
\[
3x - 2x + 5 = 2x - 2x + 9
\]
Simplifies to:
\[
x + 5 = 9
\]
2. *Subtract 5 from both sides* to move the constant term to the other side:
\[
x + 5 - 5 = 9 - 5
\]
Simplifies to:
\[
x = 4
\]
3. *Check your solution* by substituting \(x = 4\) back into the original equation:
\[
3(4) + 5 = 2(4) + 9
\]
Simplifies to:
\[
12 + 5 = 8 + 9
\]
\[
17 = 17 \quad \text{(True, so the solution is correct)}
\]
Example 2: Equation Requiring Distribution
**Equation**: \(2(3x - 4) = 3(x + 2)\)
1. *Distribute* on both sides:
\[
2 \cdot 3x - 2 \cdot 4 = 3 \cdot x + 3 \cdot 2
\]
Simplifies to:
\[
6x - 8 = 3x + 6
\]
2. **Subtract \(3x\) from both sides**:
\[
6x - 3x - 8 = 3x - 3x + 6
\]
Simplifies to:
\[
3x - 8 = 6
\]
3. **Add 8 to both sides**:
\[
3x - 8 + 8 = 6 + 8
\]
Simplifies to:
\[
3x = 14
\]
4. **Divide both sides by 3**:
\[
\frac{3x}{3} = \frac{14}{3}
\]
Simplifies to:
\[
x = \frac{14}{3} \quad \text{or approximately } 4.67
\]
5. *Check your solution* by substituting \(x = \frac{14}{3}\) back into the original equation.
Example 3: Equation with Variables on Both Sides and Like Terms
**Equation**: \(4x + 7 = 2x - 5 + 3x\)
1. **Combine like terms on the right side**:
\[
4x + 7 = 5x - 5
\]
2. **Subtract \(5x\) from both sides**:
\[
4x - 5x + 7 = 5x - 5x - 5
\]
Simplifies to:
\[
-x + 7 = -5
\]
3. **Subtract 7 from both sides**:
\[
-x + 7 - 7 = -5 - 7
\]
Simplifies to:
\[
-x = -12
\]
4. *Multiply both sides by -1* to solve for \(x\):
\[
x = 12
\]
5. *Check your solution* by substituting \(x = 12\) back into the original equation:
\[
4(12) + 7 = 2(12) - 5 + 3(12)
\]
Simplifies to:
\[
48 + 7 = 24 - 5 + 36
\]
\[
55 = 55 \quad \text{(True, so the solution is correct)}
\]
Key Points:
- **Keep equations balanced**: Whatever operation you perform on one side, you must do to the other side.
- **Systematically simplify**: Combine like terms and use the distributive property where necessary.
- **Isolate the variable**: Move all variable terms to one side and constant terms to the other before solving.
- **Verify your solution**: Always substitute your solution back into the original equation to ensure accuracy.
By following these steps, you can effectively solve equations with variables on both sides.
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Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa
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