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Chapter 1: Problem 56

Solve each equation, and check the solution. \(\frac{3}{11} x=-5\)

Short Answer

Expert verified
The solution is \( x = -\frac{55}{3} \), and it is verified.

Step by step solution

01

- Isolate the variable

To solve for the variable, multiply both sides of the equation by the reciprocal of the coefficient of the variable. The coefficient of the variable is \( \frac{3}{11} \). Therefore, multiply both sides by \( \frac{11}{3} \): \( x = -5 \times \frac{11}{3} \).
02

- Simplify the expression

Perform the multiplication to isolate \( x \: \( x = -\frac{55}{3} \).\)
03

- Check the solution

Substitute \( x = -\frac{55}{3} \) back into the original equation to verify: \( \frac{3}{11} \times -\frac{55}{3} = -5 \). Simplify the left-hand side: \( -\frac{55}{11} = -5 \), which simplifies to \( -5 = -5 \). Since both sides of the equation are equal, the solution is verified.

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Key Concepts

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

step-by-step solution
When solving linear equations, it鈥檚 important to follow a step-by-step method to ensure you get the correct solution. The process involves breaking down the equation into simpler parts and solving for the variable systematically. Let鈥檚 dive deeper into each of these steps.

  • First, understand the equation: Identify the terms involving the variable and constants.
  • Next, isolate the variable: This means you need to perform operations to get the variable alone on one side of the equation.
  • Finally, check the solution: Substitute the obtained value back into the original equation to verify its correctness.

By following this structured approach, solving linear equations becomes manageable. It's like following a recipe to bake a cake; if you follow the steps correctly, you will get the desired result.
isolate the variable
Isolating the variable is perhaps the most crucial step in solving linear equations. To isolate the variable means to get it alone on one side of the equation. In the given exercise, we followed these steps:

  • Identify the coefficient of the variable: Here, the coefficient of x is \( \frac{3}{11} \).
  • Find the reciprocal of the coefficient: The reciprocal of \( \frac{3}{11} \) is \( \frac{11}{3} \).
  • Multiply both sides by the reciprocal: This operation helps in canceling out the coefficient, thereby isolating the variable. So, we multiply both sides of \( \frac{3}{11} x = -5 \) by \( \frac{11}{3} \). The equation becomes \( x = -5 \times \frac{11}{3} \).
  • Simplify the expression: Perform the multiplication to get \( x = -\frac{55}{3} \).

After these steps, x is successfully isolated, and we have solved for its value.
check the solution
Once you've found a potential solution, it's always good practice to check that your solution is correct. Here鈥檚 how you can check:

  • Substitute the solution back into the original equation: In this case, replace x with \( -\frac{55}{3} \) in the equation \( \frac{3}{11} x = -5 \).
  • Simplify the equation: Calculate \( \frac{3}{11} \times -\frac{55}{3} \). The left-hand side simplifies to \( -5 \).
  • Verify both sides are equal: Since \( -5 = -5 \), the left-hand side equals the right-hand side, verifying that our solution is correct.

Checking the solution ensures that there was no mistake in our calculations and provides assurance that the derived value of the variable is indeed correct. This step solidifies our confidence in the answer.

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