How do you solve the algebraic expression 8x + 4 = 4x + 3 + 4 + 6x - 4 + 4?
Understanding Algebraic Simplification
Solving an algebraic equation is much like tidying up a messy room. Before you can find the value of the unknown variable (x), you must organize the terms on both sides of the equals sign. The expression you provided is a classic example of linear equation simplification.
Step-by-Step Breakdown
To solve the equation 8x + 4 = 4x + 3 + 4 + 6x - 4 + 4, we follow a logical sequence of operations:
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Simplify the Right Side: Combine all the constant numbers and the x terms separately. On the right side, we have (4x + 6x) and (3 + 4 - 4 + 4). This simplifies to 10x + 7.
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Rewrite the Equation: Now the equation looks like this: 8x + 4 = 10x + 7.
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Isolate the Variable: Subtract 8x from both sides to keep the x terms together. This gives us 4 = 2x + 7.
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Isolate the Constant: Subtract 7 from both sides. This results in -3 = 2x.
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Solve for x: Divide both sides by 2. The final answer is x = -1.5.
Quick Reference Table
| Operation | Purpose | Example |
|---|---|---|
| Combining Like Terms | Simplifying the expression | 4x + 6x = 10x |
| Subtraction Property | Moving terms across the equals sign | 8x - 8x = 0 |
| Division Property | Isolating the variable | -3 / 2 = -1.5 |
Real-World Examples
Algebra isn't just for textbooks! Think of this as balancing a budget. If x represents the cost of an item, the equation represents two different ways to calculate a total bill. By simplifying, you are essentially finding the unit price that makes both payment methods equal.
Common Pitfalls
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Sign Errors: The most common mistake is forgetting to change the sign when moving a term across the equals sign. Always remember that if you subtract a positive number, it becomes negative on the other side.
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Incomplete Simplification: Students often try to move terms before they have fully combined all the constants on one side. Always simplify each side of the equation completely before performing any algebraic moves.