Question 2 (Essay Worth 10 Points)(07.02 MC)An Equation Is Shown Below:3(2x - 7) = 3Part A: How Many
Understanding the Given Equation
When approaching any algebraic problem, especially equations like 3(2x - 7) = 3, it’s essential to understand the structure and components involved. The equation provided is a linear equation involving a single variable, x. Our goal is to solve for x and understand the steps involved in reaching the solution.
This type of problem is fundamental in algebra because it introduces concepts such as distributive property, combining like terms, and isolating the variable to find its value. Learning to solve such equations builds a foundation for more complex algebraic manipulations used in various mathematical and real-world contexts.
Breaking Down the Equation
Before solving, let's analyze the equation:
3(2x - 7) = 3
- The left side involves a multiplication of 3 with the binomial (2x - 7).
- The right side is simply the number 3.
The goal is to find the value of x that makes this equation true.
Step 1: Apply the Distributive Property
The distributive property allows us to eliminate parentheses by multiplying each term inside the parentheses by the factor outside. For this equation:
3(2x - 7) = 3
becomes
3 2x - 3 7 = 3
which simplifies to:
6x - 21 = 3
Step 2: Isolate the Variable Term
Next, we aim to get all terms containing x on one side and constant terms on the other.
Adding 21 to both sides:
6x - 21 + 21 = 3 + 21
which simplifies to:
6x = 24
Step 3: Solve for x
To find x, divide both sides of the equation by 6:
6x / 6 = 24 / 6
which simplifies to:
x = 4
Interpreting the Solution
The value x = 4 satisfies the original equation. To verify, substitute x back into the original equation:
3(2(4) - 7) = 3
Calculate inside the parentheses:
2(4) = 8
8 - 7 = 1
Now multiply by 3:
3 1 = 3
Since both sides equal 3, the solution x = 4 is correct.
Understanding the Significance of the Solution
The process demonstrates fundamental algebraic skills:
- Distributive property application
- Combining like terms
- Isolating the variable
- Verification of the solution
These skills are critical in solving similar equations and in understanding how variables relate within equations.
Common Mistakes and How to Avoid Them
When solving equations like 3(2x - 7) = 3, students often make mistakes that hinder their success. Below are common errors and tips to avoid them:
- Failing to distribute correctly: Remember to multiply every term inside the parentheses by the factor outside. For example, 3(2x - 7) becomes 6x - 21, not just 3(2x) - 7.
- Incorrectly handling signs: Be cautious when adding or subtracting terms, especially with negative numbers.
- Dividing incorrectly: Always divide the entire side by the coefficient of x when isolating the variable.
- Forgetting to verify the solution: Substitute the found value back into the original equation to confirm accuracy.
Extensions and Related Concepts
Once comfortable with solving simple linear equations like 3(2x - 7) = 3, students can extend their skills to more complex problems, such as:
Solving Equations with Variables on Both Sides
For example:
2x + 5 = x - 3
This requires moving all variable terms to one side and constants to the other.
Working with Inequalities
Similar principles apply, but solutions are expressed as ranges rather than specific values.
Application in Word Problems
Translating real-world scenarios into equations, such as calculating costs, distances, or quantities, enhances understanding and practical skills.
Importance of Mastering Basic Algebraic Equations
Mastery of solving equations like 3(2x - 7) = 3 is fundamental for progressing in mathematics. It develops critical thinking, problem-solving skills, and the ability to analyze data. These skills are not only essential academically but also valuable in everyday decision-making and various professional fields.
Conclusion
In summary, solving the equation 3(2x - 7) = 3 involves understanding the distributive property, combining like terms, isolating the variable, and verifying the solution. The process illustrates key algebraic techniques, emphasizing the importance of careful calculations and logical reasoning. By mastering these foundational skills, students can confidently approach more complex equations and mathematical concepts, laying a solid groundwork for future learning and application.
Whether you’re preparing for tests, working on homework, or seeking to improve your problem-solving skills, understanding how to solve equations like this is a vital step in your mathematical journey.
