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Chapter 2: Problem 125

Solve each equation. Be sure to check each answer. $$\frac{1}{7} x=\frac{5}{7}$$

Short Answer

Expert verified
The solution to the equation \(\frac{1}{7}x = \frac{5}{7}\) is \(x = 5\).

Step by step solution

01

Isolate the variable 'x'

The equation \(\frac{1}{7}x = \frac{5}{7}\) can be solved for x by multiplying each side of the equation by 7. This gives \(x = 5\).
02

Check the solution

Check the solution by substituting x = 5 back into the original equation. This gives \(\frac{1}{7}*5 = \frac{5}{7}\), which simplifies to \(\frac{5}{7} = \frac{5}{7}\). The two sides of the equation match, meaning that the solution x = 5 is correct.

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Key Concepts

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

Isolate the Variable
Breaking down complex equations into simpler forms is a key aspect of solving algebraic problems. To isolate the variable means to get the variable on one side of the equation all by itself. In the case of \( \frac{1}{7}x = \frac{5}{7} \), we want to isolate X. This is achieved by performing the same operation on both sides of the equation. Since x is being multiplied by \( \frac{1}{7} \), we do the opposite operation 鈥� we multiply both sides by the reciprocal of \( \frac{1}{7} \), which is 7. Here鈥檚 how it鈥檚 done stepwise:

  • Multiply both sides of the original equation \( \frac{1}{7}x = \frac{5}{7} \) by 7.
  • The sevens on the left side cancel out, leaving us with just x on the left side.
  • On the right side, multiplying 7 by \( \frac{5}{7} \) leaves us with 5.
  • The equation now reads x = 5, which is the simplified form with the variable isolated.
Checking Solutions in Algebra
After finding the potential solution to an equation, it鈥檚 crucial to verify its accuracy. Checking the solution safeguards against errors made in the process of solving the equation. To check the solution in algebra, substitute the value back into the original equation and ensure each side balances. For our example, we determined that x = 5. To check:

  • Substitute x with 5 in \( \frac{1}{7}x = \frac{5}{7} \).
  • Simplify the left side to find \( \frac{5}{7} \).
  • Compare the simplified left side, \( \frac{5}{7} \), with the original right side of the equation.
  • If they match, as they do in this case with both sides equal to \( \frac{5}{7} \), the solution is correct.
It is essential to ensure the equation holds true for the found value. If it does not, it indicates a mistake was made somewhere in the process. Checking your solutions is a marker of good mathematical practice and understanding.
Linear Equation Step-by-Step Solution
Solving a linear equation involves finding the value of the variable that makes the equation true. Here鈥檚 a step-by-step breakdown of solving our given equation \( \frac{1}{7}x = \frac{5}{7} \):

  • Identify the operations involved. In this case, x is being multiplied by \( \frac{1}{7} \).
  • Perform the opposite operation to isolate x. We multiply each side by 7: \( 7 \cdot \frac{1}{7}x = 7 \cdot \frac{5}{7} \).
  • Simplify the equation. Multiplication by 7 cancels out the fraction on the left, leaving x by itself: x = 5.
  • Verify your solution by plugging it back into the original equation.
  • Confirm that the left and right sides of the equation are equal after the substitution, verifying that the solution is correct.
Following these steps carefully, without skipping any, ensures a methodical approach to solving linear equations. Such a structured method not only helps in solving the equations correctly but also builds a strong foundation for tackling more complex problems in the future.

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Most popular questions from this chapter

The perimeter \(P\) of a rectangle with length \(l\) and width \(w\) is given by the formula \(P=2 l+2 w .\) Solve this formula for \(w .\) If the perimeter of a certain rectangle is 58.37 meters and its length is 17.23 meters, find its width. Round to two decimal places.

Solve each inequality and graph the solution on the number line. $$-4 < \frac{x-2}{2} < 6$$

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The outside perimeter of the triangular brace shown in the illustration is 57 feet. If all three sides are of equal length, find the length of each side.

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