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Chapter 11: Problem 5

Solve. Clear fractions first. $$ \frac{3}{7} x-x=1 $$

Short Answer

Expert verified
x = -\frac{7}{4}

Step by step solution

01

- Eliminate Fractions

To eliminate the fraction \(\frac{3}{7}x\), we will multiply each term of the equation by 7, the denominator of the fraction. This will clear the fraction. \[ 7 \times \frac{3}{7}x - 7 \times x = 7 \times 1 \] Simplifying, we get: \[ 3x - 7x = 7 \]
02

- Combine Like Terms

Combine like terms on the left side of the equation. \[ 3x - 7x = 7 \] Simplify the left side: \[ -4x = 7 \]
03

- Solve for x

Isolate the variable \(x\) by dividing both sides of the equation by -4. \[ x = \frac{7}{-4} \] Simplify the fraction (if needed). Since \(\frac{7}{-4}\) cannot be further simplified, we keep it as is.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Eliminating Fractions
When solving linear equations, fractions can often complicate the process. The first step to simplify things is to eliminate any fractions. In our problem, we have the equation: \(\frac{3}{7} x - x = 1\). To eliminate the fraction \(\frac{3}{7}x\), you should multiply every term in the equation by the denominator of the fraction, which is 7. Here鈥檚 how it looks: \[ 7 \times \frac{3}{7}x - 7 \times x = 7 \times 1 \] This multiplication clears the fraction, as multiplying \(\frac{3}{7}\) by 7 gives us 3. So, the equation simplifies to: \[ 3x - 7x = 7 \] This step makes the equation easier by removing the fraction, leaving you with simpler numbers to work with.
Combining Like Terms
After clearing the fractions, the next step is to combine like terms. In our simplified equation: \(\begin{align*} 3x - 7x = 7, \end{align*}\), the terms \(3x\) and \(-7x\) are like terms because they both contain the variable \((x)\). Combining them means adding or subtracting them according to their coefficients. \[ 3x - 7x = (3 - 7)x = -4x \] Now the equation is simplified further: \[ -4x = 7 \] The goal here is to get the equation to its simplest form before isolating the variable.
Isolating the Variable
The final step in solving the equation is to isolate the variable, which means getting \(x\) by itself on one side of the equation. In the current simplified form: \[ -4x = 7, \] To isolate \(x\), you need to divide both sides of the equation by the coefficient of \(x\), which is -4. So, divide both sides by -4: \[ x = \frac{7}{-4} \] Simplify the fraction if possible. However, \(\frac{7}{-4}\) cannot be simplified further, so the solution is: \[ x = \frac{7}{-4} \] The variable \(x\) is now isolated, and you have successfully solved the equation. Remember, isolating the variable is about using inverse operations to get \(x\) alone on one side of the equation.

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