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Chapter 6: Problem 10

Solve each equation. $$ \frac{x-7}{x-1}=\frac{11}{x-1} $$

Short Answer

Expert verified
The solution to the equation is \( x = 18 \).

Step by step solution

01

Cross-Multiply

Since both sides of the equation \( \frac{x-7}{x-1} = \frac{11}{x-1} \) have the same denominator, we can multiply both sides by \(x-1\) to eliminate the denominator. This gives us the equation:\[(x-7) = 11\].
02

Solve for x

Now that we have the simplified equation \( x - 7 = 11 \), we can solve for \( x \) by adding \( 7 \) to both sides of the equation:\[ x - 7 + 7 = 11 + 7 \]\[ x = 18 \].
03

Verify the Solution

Substitute \( x = 18 \) back into the original equation to ensure it holds true:\[ \frac{18-7}{18-1} = \frac{11}{18-1} \]\[ \frac{11}{17} = \frac{11}{17} \]Since the left side equals the right side, \( x = 18 \) is the correct solution.

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Key Concepts

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

Cross-Multiplication Method
When dealing with rational equations, the cross-multiplication method is a powerful technique that helps to simplify and solve them effortlessly. In the given problem, the equation \( \frac{x-7}{x-1} = \frac{11}{x-1} \) presents a classic scenario where cross-multiplication can be applied. Since both sides of the equation have the same denominator \( x-1 \), we can safely eliminate it.
  • Eliminate the denominator: By multiplying both sides by \( x-1 \), we remove the rational component, making the equation easier to handle: \[(x-7) = 11\].
  • Simplify: Simplification removes complexity and brings the equation to a familiar, more manageable form, which is crucial for solving it.
This step transforms the equation into a simple linear equation, paving the way to find the solution quickly. Cross-multiplying is advantageous when the denominators on both sides of the equation are identical or can be made identical.
Solving Equations
Solving equations is the process of finding the value of the variable that makes the equation true. Once the equation is simplified through cross-multiplication, solving it becomes straightforward.
  • Identifying the operation: In the linear equation \( x - 7 = 11 \), we must determine the operation needed to isolate \( x \).
  • Undoing the operation: To isolate \( x \), we need to undo the subtraction of 7. Thus, we add 7 to both sides:\[x - 7 + 7 = 11 + 7\].
  • Finding the solution: This results in \( x = 18 \).
The systematic approach of undoing operations ensures you follow logical steps to find the variable's value. Understanding each operation's purpose helps you tackle even more complex equations confidently.
Verifying Solutions
Verification is a crucial step in the process of solving equations. It confirms that the solution is correct and satisfies the original equation. This step ensures the solution's accuracy and provides peace of mind.
  • Substitute back: Replace \( x \) in the original equation with the solution you found, \( x = 18 \):\[ \frac{18-7}{18-1} = \frac{11}{18-1} \].
  • Check equality: Simplify both sides to verify equality:\[ \frac{11}{17} = \frac{11}{17} \].
  • Confirm solution: Since both sides are equal, it verifies \( x = 18 \) is the correct solution.
Verification is the final checkpoint in solving equations, ensuring there were no algebraic errors made during calculation. It is always wise to make this a regular practice, reinforcing problem-solving accuracy.

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