Solving One-Step and Two-Step Equations
Key Concepts
1. **One-Step Equations**: Equations that require only one operation to solve. For example, \( x + 3 = 7 \).
2. **Two-Step Equations**: Equations that require two operations to solve. For example, \( 2x + 3 = 7 \).
Detailed Explanation
One-Step Equations
To solve one-step equations, isolate the variable by performing the inverse operation. For example, if the equation is \( x + 3 = 7 \), subtract 3 from both sides to isolate \( x \):
\[ x + 3 - 3 = 7 - 3 \]
\[ x = 4 \]
Two-Step Equations
To solve two-step equations, first isolate the term with the variable by performing the inverse operation of the constant term. Then, solve for the variable by performing the inverse operation of the coefficient. For example, if the equation is \( 2x + 3 = 7 \):
1. Subtract 3 from both sides:
\[ 2x + 3 - 3 = 7 - 3 \]
\[ 2x = 4 \]
2. Divide both sides by 2:
\[ \frac{2x}{2} = \frac{4}{2} \]
\[ x = 2 \]
Examples
Example 1: Solve \( x - 5 = 10 \).
Solution: Add 5 to both sides:
\[ x - 5 + 5 = 10 + 5 \]
\[ x = 15 \]
Example 2: Solve \( 3x - 2 = 10 \).
Solution: 1. Add 2 to both sides:
\[ 3x - 2 + 2 = 10 + 2 \]
\[ 3x = 12 \]
2. Divide both sides by 3:
\[ \frac{3x}{3} = \frac{12}{3} \]
\[ x = 4 \]
Analogies
Think of solving equations as balancing a scale. To keep the scale balanced, whatever operation you perform on one side, you must perform on the other side. For one-step equations, you are just adding or removing weights from one side. For two-step equations, you first adjust the weights and then find the correct balance point.
Practical Application
Understanding how to solve one-step and two-step equations is crucial in various real-world scenarios, such as calculating costs, determining quantities, and solving problems in physics and economics. Mastering these operations ensures accurate and reliable results in any application.