-8 + 2x - 9 + 3x - 10 = 21
Solving algebraic equations like -8 + 2x - 9 + 3x - 10 = 21 is a fundamental skill in mathematics that forms the basis for more advanced topics. Whether you're a student brushing up on algebra or someone interested in understanding how to manipulate equations, this article offers a comprehensive guide to solving such equations efficiently. We'll explore the step-by-step process, explain relevant concepts, and provide tips to master similar algebraic problems.
---
Understanding the Equation: Breaking Down the Components
Before diving into solving the equation, it's essential to understand its components:
- Constants: Numbers without variables, such as -8, -9, and -10.
- Variables: Symbols representing unknown quantities, in this case, x.
- Coefficients: Numbers multiplying variables, like 2 and 3 in 2x and 3x.
The given equation:
\[
-8 + 2x - 9 + 3x - 10 = 21
\]
contains multiple constants and variables. To solve it, we aim to isolate x on one side of the equation.
---
Step-by-Step Solution Process
The process involves several systematic steps:
- Combine like terms on the left side.
- Simplify the equation to isolate the term with x.
- Solve for the variable x.
- Verify the solution by substituting back into the original equation.
Let's proceed with these steps in detail.
---
Step 1: Combine Like Terms
The left side contains constants: -8, -9, and -10, and variables: 2x and 3x.
- Constants: \(-8 - 9 - 10\)
- Variables: \(2x + 3x\)
Calculating:
- Constants: \(-8 - 9 = -17\); then \(-17 - 10 = -27\)
- Variables: \(2x + 3x = 5x\)
Now, the equation simplifies to:
\[
-27 + 5x = 21
\]
---
Step 2: Isolate the Variable Term
To solve for x, first move the constant term to the other side:
\[
5x = 21 + 27
\]
Adding \(27\) to both sides:
\[
5x = 48
\]
---
Step 3: Solve for x
Divide both sides by the coefficient of x, which is 5:
\[
x = \frac{48}{5}
\]
Expressed as a mixed number or decimal:
- As a decimal: \(x = 9.6\)
- As a fraction: \(\frac{48}{5}\)
---
Step 4: Verify the Solution
Substitute \(x = \frac{48}{5}\) back into the original equation:
\[
-8 + 2x - 9 + 3x - 10 = 21
\]
Calculate each term:
- \(2x = 2 \times \frac{48}{5} = \frac{96}{5}\)
- \(3x = 3 \times \frac{48}{5} = \frac{144}{5}\)
Rewrite the entire equation with common denominators:
\[
-8 - 9 - 10 + \frac{96}{5} + \frac{144}{5} = 21
\]
Express constants as fractions with denominator 5:
\[
-8 = -\frac{40}{5}
\]
\[
-9 = -\frac{45}{5}
\]
\[
-10 = -\frac{50}{5}
\]
Sum constants:
\[
-\frac{40}{5} - \frac{45}{5} - \frac{50}{5} = -\frac{135}{5} = -27
\]
Sum variable terms:
\[
\frac{96}{5} + \frac{144}{5} = \frac{240}{5} = 48
\]
Now, sum all:
\[
-27 + 48 = 21
\]
Since the left side equals 21, which matches the right side, the solution is confirmed.
---
Additional Tips for Solving Similar Equations
- Combine like terms early: Always simplify constants and variables first to make the equation easier to work with.
- Maintain balance: Whatever operation you perform on one side, do the same on the other.
- Check your work: Substituting your solution back into the original equation can prevent errors.
- Practice with different equations: The more you practice, the more intuitive algebra becomes.
Common Mistakes to Avoid
- Ignoring signs: Be careful with negative signs when combining terms.
- Forgetting to distribute: When equations involve parentheses, distribute correctly before simplifying.
- Incorrectly combining like terms: Only combine terms with the same variable and degree.
- Overlooking the solution verification: Always verify your answer to confirm accuracy.
Real-World Applications of Solving Equations
Algebraic equations like the one discussed are not just mathematical exercises—they have real-world applications:
- Finance: Calculating interest, loans, and investments.
- Science: Determining unknown quantities in experiments.
- Engineering: Designing systems based on variable relationships.
- Computer Programming: Solving for unknowns in algorithms and data analysis.
Understanding how to manipulate and solve equations enhances problem-solving skills applicable across various fields.
---
Summary
In summary, solving the equation -8 + 2x - 9 + 3x - 10 = 21 involves:
- Combining like terms to simplify the equation.
- Isolating the variable by performing inverse operations.
- Calculating the value of x.
- Verifying the solution to ensure correctness.
The final answer, x = \(\frac{48}{5}\) or 9.6, demonstrates the power of algebraic techniques to find unknown quantities systematically.
---
Further Practice Problems
To strengthen your algebra skills, try solving these similar equations:
- 3x + 7 - 2x = 10
- -5 + 4x - 3 + 2x = 9
- 2(3x - 4) = 14
- 6x + 3 = 2x + 15
- -8 + 2x - 9 + 3x - 10 = 21 (the original problem)
Practicing these problems will help you become more comfortable with algebraic manipulation and problem-solving.
---
In conclusion, mastering the process of solving equations like -8 + 2x - 9 + 3x - 10 = 21 is vital for progressing in mathematics. By understanding each step, practicing regularly, and verifying solutions, you can confidently tackle a wide range of algebraic problems.
