Solving the Equation 5(7x + 15) = 9(14x + 9): A Step-by-Step Guide
Understanding how to solve linear equations with parentheses is a fundamental algebra skill that forms the foundation for more advanced mathematical concepts. The equation 5(7x + 15) = 9(14x + 9) demonstrates key techniques including distribution, combining like terms, and isolating variables. Mastering these methods will help you tackle similar problems with confidence and precision.
Introduction to Linear Equations with Parentheses
Linear equations often appear in forms that require careful manipulation before the solution becomes apparent. When parentheses are involved, the first step typically involves applying the distributive property to eliminate them. This process transforms complex-looking expressions into simpler forms that can be solved using basic arithmetic operations No workaround needed..
The equation 5(7x + 15) = 9(14x + 9) contains two sets of parentheses, each multiplied by a coefficient. To solve for x, we need to systematically simplify both sides and then isolate the variable.
Step-by-Step Solution Process
Step 1: Apply the Distributive Property
Begin by distributing the coefficients outside each set of parentheses:
Left side: 5(7x + 15)
- Multiply 5 by 7x: 5 × 7x = 35x
- Multiply 5 by 15: 5 × 15 = 75
- Result: 35x + 75
Right side: 9(14x + 9)
- Multiply 9 by 14x: 9 × 14x = 126x
- Multiply 9 by 9: 9 × 9 = 81
- Result: 126x + 81
After distribution, the equation becomes: 35x + 75 = 126x + 81
Step 2: Move All Variable Terms to One Side
To isolate x, we need to gather all terms containing x on one side and constant terms on the other. Let's move the variable terms to the left side:
Subtract 126x from both sides: 35x - 126x + 75 = 126x - 126x + 81 -91x + 75 = 81
Step 3: Move Constant Terms to the Opposite Side
Next, move the constant term (75) to the right side by subtracting 75 from both sides:
-91x + 75 - 75 = 81 - 75 -91x = 6
Step 4: Solve for x
Now divide both sides by -91 to isolate x:
x = 6 ÷ (-91) x = -6/91
Verification of the Solution
It's essential to verify our answer by substituting it back into the original equation:
Left side: 5(7x + 15) = 5(7(-6/91) + 15) = 5(-42/91 + 15) = 5(-42/91 + 1365/91) = 5(1323/91) = 6615/91 = 72.69.. The details matter here..
Right side: 9(14x + 9) = 9(14(-6/91) + 9) = 9(-84/91 + 9) = 9(-84/91 + 819/91) = 9(735/91) = 6615/91 = 72.69.. It's one of those things that adds up..
Since both sides equal the same value, our solution is correct.
Alternative Approach: Collecting Terms Differently
While the previous method works perfectly, some students prefer moving terms in the opposite direction. Starting from: 35x + 75 = 126x + 81
We could subtract 35x and 81 from both sides: 75 - 81 = 126x - 35x -6 = 91x x = -6/91
This confirms our answer using a slightly different approach No workaround needed..
Common Mistakes to Avoid
When solving equations like 5(7x + 15) = 9(14x + 9), students often make these errors:
- Incorrect distribution: Forgetting to multiply the coefficient by every term inside the parentheses
- Sign errors: Mismanaging negative signs when moving terms across the equals sign
- Arithmetic mistakes: Making calculation errors when combining like terms
- Fraction simplification: Not reducing fractions to their simplest form
Always double-check each step, especially when dealing with negative numbers and fractions.
Scientific Explanation of the Mathematical Principles
The solution relies on several fundamental algebraic principles:
- Distributive Property: a(b + c) = ab + ac
- Equality Preservation: Performing identical operations on both sides maintains equality
- Inverse Operations: Using addition/subtraction and multiplication/division to isolate variables
- Combining Like Terms: Grouping similar terms to simplify expressions
These principles work together to transform complex equations into simple solutions Simple as that..
Practical Applications
Linear equations like this appear in various real-world contexts:
- Financial calculations involving interest rates
- Physics problems related to motion and forces
- Engineering applications requiring proportional relationships
- Business scenarios involving cost and revenue analysis
Understanding how to solve them efficiently is crucial for academic success and practical problem-solving.
Frequently Asked Questions
Q: Can this equation be solved without distributing first? A: While possible by dividing both sides by common factors, distribution is typically the most straightforward approach Surprisingly effective..
Q: Is -6/91 the simplest form of the answer? A: Yes, since 6 and 91 share no common factors other than 1 (91 = 7 × 13).
Q: What would happen if the coefficients were larger? A: The process remains identical; only the arithmetic becomes more complex.
Q: How can I check my work more efficiently? A: Substitute your answer into the original equation using a calculator for decimal verification That alone is useful..
Conclusion
Solving 5(7x + 15) = 9(14x + 9) requires systematic application of algebraic principles: distribution, term collection, and variable isolation. The solution x = -6/91 demonstrates how even seemingly complex equations can be resolved through methodical steps Practical, not theoretical..
Mastering these techniques builds confidence for tackling more advanced mathematical challenges. Remember to verify your solutions and practice regularly to develop fluency with linear equations. With consistent practice, problems that initially seem intimidating become routine exercises in logical reasoning That's the whole idea..
