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Chapter 2: Problem 59

Solve each equation. Check each solution. $$ 10-3 x-6-9 x=7 $$

Short Answer

Expert verified
The solution is \( x = -\frac{1}{4} \).

Step by step solution

01

Simplify the Equation

First, combine like terms on the left side of the equation. The terms with constants are 10 and -6, and the terms with 'x' are -3x and -9x.
02

Combine Constants

Calculate \(10 - 6 = 4\). The new equation is \(4 - 3x - 9x = 7\).
03

Combine X-Terms

Combine the 'x' terms: \(-3x - 9x = -12x\). The equation becomes \(4 - 12x = 7\).
04

Isolate the Variable

Subtract 4 from both sides to isolate the term with 'x': \( -12x = 7 - 4 \). Simplifying gives \(-12x = 3\).
05

Solve for x

Divide both sides by -12 to solve for 'x': \( x = \frac{3}{-12} \). This simplifies to \( x = -\frac{1}{4} \).
06

Verify the Solution

Substitute \( x = -\frac{1}{4} \) back into the original equation: \( 10 - 3\left(-\frac{1}{4}\right) - 6 - 9\left(-\frac{1}{4}\right) \). Simplify each term: \(10 + \frac{3}{4} - 6 + \frac{9}{4}\). \(4 + \frac{12}{4} = 4 + 3 = 7\). The left side equals the right side of the original equation, verifying \( x = -\frac{1}{4} \) is correct.

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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, one important step is combining like terms. Like terms are those that have the same variable part or no variable at all. For example, in the equation \(10 - 3x - 6 - 9x = 7\), the terms \(10\) and \(-6\) are like terms because they are constants (numbers without any variables). Similarly, \(-3x\) and \(-9x\) are like terms because they both include the variable \(x\).

To combine like terms:
  • Group the constants together.
  • Group the terms with the same variables together.
By simplifying, \(10 - 6\) becomes \(4\), and \(-3x - 9x\) simplifies to \(-12x\). After combining these terms, our equation reduces to \(4 - 12x = 7\). This step makes equations more manageable and prepares them for the next stage of solving.
Isolating Variables
The goal when solving an equation is to find out what value the variable (usually represented by \(x\)) has. To do this, you need to isolate the variable on one side of the equation. In our exercise, after combining terms, we had: \(4 - 12x = 7\). The plan is to get \(x\) by itself on one side.

Here's how to isolate a variable:
  • Move other terms to the opposite side by performing arithmetic operations. For this equation, subtract 4 from both sides to get \(-12x = 3\).
  • Divide both sides by the coefficient of \(x\), which is \(-12\). This simplifies our equation to \(x = \frac{3}{-12}\), which simplifies further to \(x = -\frac{1}{4}\).
Once the variable is isolated, you've effectively solved for \(x\). The value here indicates what \(x\) should be so that the original equation holds true.
Verifying Solutions
After solving for the variable, it's crucial to verify the solution. This ensures that the answer is correct and that the original equation is satisfied. To verify a solution, simply substitute the value back into the original equation and see if it holds true.

For our equation, substitute \(x = -\frac{1}{4}\) back in:
  • Replace every \(x\) with \(-\frac{1}{4}\) in the original equation \(10 - 3x - 6 - 9x = 7\).
  • Calculate each term: \(10 - 3\left(-\frac{1}{4}\right) - 6 - 9\left(-\frac{1}{4}\right)\).
  • This simplifies to \(10 + \frac{3}{4} - 6 + \frac{9}{4}\), which further simplifies to \(4 + \frac{12}{4}\).
  • The final value equals \(7\), which matches the right side of the original equation.
Thus, the solution \(x = -\frac{1}{4}\) is verified correctly. Verifying solutions helps build confidence in the accuracy of your calculations and reassures you that the process was followed correctly.

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