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Magazine Chapter 2: Problem 14
Solve each equation and check. $$ 13 x-15 x+8=4 x+2-24 $$
Short Answer
Expert verified
The solution is \(x = 5\).
Step by step solution
01
Simplify Both Sides
Combine like terms on each side of the equation. On the left side, combine \(13x\) and \(-15x\) to get \(-2x\). The equation now is: \[-2x + 8 = 4x + 2 - 24\]On the right side, combine \(2\) and \(-24\) to get \(-22\). The equation becomes: \[-2x + 8 = 4x - 22\]
02
Isolate the Variable
To get all \(x\) terms on one side of the equation, add \(2x\) to both sides: \[8 = 6x - 22\]Now move constant terms to the other side by adding \(22\) to both sides of the equation: \[30 = 6x\]
03
Solve for x
To solve for \(x\), divide both sides by \(6\): \[x = \frac{30}{6} = 5\]
04
Verify the Solution
Substitute \(x = 5\) back into the original equation to ensure the solution is correct. The original equation is: \[13(5) - 15(5) + 8 = 4(5) + 2 - 24\]Calculate each side:- Left Side: \[(65) - (75) + 8 = -10 + 8 = -2\]- Right Side: \[20 + 2 - 24 = 22 - 24 = -2\]Both sides are equal, therefore the solution \(x = 5\) is verified.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Combining Like Terms
When solving equations, it's important to start by simplifying both sides. This often involves combining like terms. Like terms are terms in an equation that have the same variable raised to the same power. For example, in the expression \(13x - 15x\), both terms have the variable \(x\) with the same exponent.
- Identify like terms: Look for terms that have the same variable parts.
- Combine them: Add or subtract their coefficients. For our exercise, \(13x - 15x\) becomes \(-2x\). On the right, numbers without variables, such as \(2\) and \(-24\), can be combined to get \(-22\).
Isolating the Variable
Once the equation is simplified, the next goal is to isolate the variable. This means rearranging the equation so that the variable is on one side by itself.
- Move variable terms: Start by moving terms with the variable to one side of the equation. In the example, adding \(2x\) to both sides simplifies the equation to \(8 = 6x - 22\).
- Eliminate constant terms: To completely isolate the variable term, move constant terms to the opposite side. Add \(22\) to both sides to simplify the equation further to \(30 = 6x\).
Verifying Solutions
Verification is an essential part of equation solving. After finding a solution, it鈥檚 important to check that it satisfies the original equation. Here's how you can verify:
- Substitute the solution back into the original equation: Replace the variable with the found value. Here, substituting \(x = 5\) into the original equation.
- Calculate both sides: Simplify each side independently. Both sides must equal for the solution to be correct.
Simplifying Expressions
Simplifying expressions is a critical step that helps make equations easier to handle. Simplification involves reducing the complexity of expressions by applying arithmetic operations and following algebraic rules to make them easier to solve.
- Combine numbers and like terms: Just as we combined like terms earlier, simplifying an expression also involves combining numerical constants.
- Apply arithmetic operations: Perform straightforward additions, subtractions, multiplications, or divisions to simplify expressions. In this exercise, once the terms were combined, simplifying resulted in equations like \(-2x + 8 = 4x - 22\).
