Understanding Two-Step Equations
Two-step equations are algebraic expressions that require two operations to isolate the variable. The general form of a two-step equation can be expressed as:
\[ ax + b = c \]
where:
- \( a \) is the coefficient of the variable \( x \),
- \( b \) is a constant,
- \( c \) is the result of the operation.
To solve a two-step equation, one generally follows these steps:
1. Eliminate the constant term by performing the inverse operation.
2. Isolate the variable by dividing or multiplying as needed.
For example, consider the equation:
\[ 2x + 5 = 15 \]
To solve this:
1. Subtract 5 from both sides:
\[ 2x = 10 \]
2. Divide both sides by 2:
\[ x = 5 \]
This systematic approach is critical for success in algebra and related fields.
Multiple-Choice Questions on Two-Step Equations
To test comprehension of two-step equations, multiple-choice questions can be an effective tool. Below are several MCQs that cover various aspects of solving two-step equations.
Question Set 1: Basic Two-Step Equations
1. Solve the equation \( 3x - 7 = 5 \).
- A) \( x = 4 \)
- B) \( x = 3 \)
- C) \( x = 2 \)
- D) \( x = 5 \)
2. What is the value of \( x \) in the equation \( 4(x + 2) = 20 \)?
- A) \( x = 5 \)
- B) \( x = 3 \)
- C) \( x = 2 \)
- D) \( x = 6 \)
3. If \( 5x + 10 = 35 \), what is \( x \)?
- A) \( x = 2 \)
- B) \( x = 5 \)
- C) \( x = 3 \)
- D) \( x = 4 \)
Question Set 2: Application of Two-Step Equations
4. A number is multiplied by 3 and then decreased by 4, resulting in 11. What is the number?
- A) 5
- B) 7
- C) 10
- D) 3
5. In the equation \( 6 = 2x + 4 \), what is the value of \( x \)?
- A) \( x = 1 \)
- B) \( x = 2 \)
- C) \( x = 3 \)
- D) \( x = 4 \)
6. Solve for \( y \): \( 2y - 3 = 11 \).
- A) \( y = 8 \)
- B) \( y = 7 \)
- C) \( y = 6 \)
- D) \( y = 5 \)
Question Set 3: Challenging Two-Step Equations
7. What is the solution for \( -3(x - 2) = 9 \)?
- A) \( x = 3 \)
- B) \( x = 5 \)
- C) \( x = 0 \)
- D) \( x = 6 \)
8. If \( 7 - 2x = 1 \), find \( x \).
- A) \( x = 2 \)
- B) \( x = 3 \)
- C) \( x = 4 \)
- D) \( x = 1 \)
9. The equation \( 10 = 5 - 2x \) needs to be solved for \( x \). What is \( x \)?
- A) \( x = 1 \)
- B) \( x = -1 \)
- C) \( x = 2 \)
- D) \( x = -2 \)
Answer Key for Two-Step Equations MCQs
Here is the answer key to the MCQs provided above, along with brief explanations for each solution.
Answers for Question Set 1
1. A) \( x = 4 \)
- Explanation: Add 7 to both sides: \( 3x = 12 \). Then divide by 3: \( x = 4 \).
2. A) \( x = 3 \)
- Explanation: Divide both sides by 4: \( x + 2 = 5 \). Subtract 2: \( x = 3 \).
3. B) \( x = 5 \)
- Explanation: Subtract 10 from both sides: \( 5x = 25 \). Divide by 5: \( x = 5 \).
Answers for Question Set 2
4. B) 7
- Explanation: Add 4 to both sides: \( 3x = 15 \). Divide by 3: \( x = 5 \).
5. B) \( x = 1 \)
- Explanation: Subtract 4: \( 2x = 2 \). Divide by 2: \( x = 1 \).
6. A) \( y = 8 \)
- Explanation: Add 3: \( 2y = 14 \). Divide by 2: \( y = 7 \).
Answers for Question Set 3
7. B) \( x = 5 \)
- Explanation: Distribute: \( -3x + 6 = 9 \). Subtract 6: \( -3x = 3 \). Divide by -3: \( x = -1 \).
8. A) \( x = 3 \)
- Explanation: Subtract 7: \( -2x = -6 \). Divide by -2: \( x = 3 \).
9. C) \( x = 2 \)
- Explanation: Rearranging gives \( -2x = -5 \). Divide by -2: \( x = 2 \).
Conclusion
A solid understanding of two-step equations is vital for anyone studying mathematics. The multiple-choice questions and answer key provided in this article serve as a valuable tool for both assessment and practice. Students can utilize these resources to enhance their problem-solving abilities and gain confidence in their mathematical skills. Mastery of two-step equations not only aids in academic success but also lays the groundwork for advanced topics in algebra and beyond. As students continue to practice, they will find that these foundational skills will serve them well throughout their educational journeys.
Frequently Asked Questions
What is a two-step equation?
A two-step equation is an algebraic equation that requires two operations to solve for the variable.
Which of the following is an example of a two-step equation?
2x + 3 = 11 is an example of a two-step equation.
What is the first step in solving the equation 3x - 5 = 7?
The first step is to add 5 to both sides of the equation.
How do you isolate the variable in the equation 2x + 4 = 12?
First, subtract 4 from both sides, then divide by 2.
What is the solution to the two-step equation 5x = 25?
The solution is x = 5.
In a multiple-choice question format, how should the answer key for a two-step equation be structured?
The answer key should list each question with its corresponding correct answer clearly identified.
