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Chapter 3: Problem 98

Solve the equation and check your solution. $$\frac{w}{3}=\frac{8}{12}$$

Short Answer

Expert verified
The solution to the equation \(\frac{w}{3} = \frac{8}{12}\) is \(w = 2\). The check also confirms this solution is correct.

Step by step solution

01

Isolate the Variable

The goal is to isolate the variable \(w\) on one side of the equation. The equation given is \(\frac{w}{3} = \frac{8}{12}\). To get \(w\) alone on one side, we can multiply both sides of the equation by 3. This will give \(w = 3 \times \frac{8}{12}\).
02

Solve the Equation

Now we can simply calculate the right side of the equation. Which gives, \(w = 3 \times \frac{2}{3}\).
03

Final Solution

Calculating the above expression, we get \(w = 2\).
04

Check the Solution

To check if the solution is correct, replace \(w\) with the calculated value in the original equation. The equation \(\frac{w}{3} = \frac{8}{12}\) becomes \(\frac{2}{3} = \frac{8}{12}\). Simplifying both sides, \(\frac{2}{3} = \frac{2}{3}\), shows that our solution is correct.

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Key Concepts

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

Isolating Variables
When solving algebraic equations, isolating the variable is often the first and most crucial step. This means you want to have the variable on its own on one side of the equation. It's like playing a game of balance. Whatever you do to one side of the equation, you have to do to the other side as well. This keeps the equation balanced, just like a seesaw.
For instance, given \[\frac{w}{3} = \frac{8}{12}\] to isolate \(w\), you multiply both sides by 3. This allows \(w\) to "break free" from being divided by 3:
  • Start with the equation: \(\frac{w}{3} = \frac{8}{12}\)
  • Multiply both sides by 3: \(3 \times \frac{w}{3} = 3 \times \frac{8}{12}\)
  • This simplifies to: \(w = 3 \times \frac{8}{12}\)
By doing this, you effectively "isolate" \(w\), getting it ready for further calculations.
Checking Solutions
Once you've found a solution for your equation, it's not enough to just put it down and move on. You should always check to ensure that your solution is correct. Why? Because checking helps to confirm your work and verify that no mistakes were made along the way.
You can check a solution by substituting the value back into the original equation. For the above exercise, where we found that \(w = 2\), you need to see if this makes the original equation true:
  • Start with the original equation: \(\frac{w}{3} = \frac{8}{12}\)
  • Substitute \(w = 2\) into the equation: \(\frac{2}{3} = \frac{8}{12}\)
  • Simplify both sides to see if they are equal: \(\frac{2}{3} = \frac{2}{3}\)
Performing the check confirms that the solution is indeed correct. This process is vital as it acts as a verification step to reassure accuracy.
Multiplying Fractions
Multiplying fractions is an essential skill, especially when dealing with algebraic equations. When you multiply fractions, you multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together.
For example, in the equation:\[w = 3 \times \frac{8}{12}\] of our exercise, you're multiplying a whole number by a fraction:
  • The whole number 3 can be written as a fraction: \(\frac{3}{1}\)
  • Multiply the numerators: \(3 \times 8 = 24\)
  • Multiply the denominators: \(1 \times 12 = 12\)
  • The result is: \(\frac{24}{12}\)
  • Simplify the fraction to: \(2\)
The simplification step is crucial because it gives you the final answer in the simplest form, making it easier to interpret in the context of the problem. Understanding how to multiply and simplify fractions will make solving algebraic equations much easier.

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Most popular questions from this chapter

Solve and graph the inequality. $$\frac{2}{3} x \leq 12$$

Solve and graph the inequality. $$18-y \leq 5$$

Solve the proportion. $$\frac{x}{3}=\frac{28}{12}$$

Error Analysis Describe and correct the error in graphing \(x<5\).

Determine whether each value of \(x\) is a solution of the inequality. \(3(x+5)-4>2\) (a) \(x=3\) (b) \(x=0\) (c) \(x=-4\) (d) \(x=-10\)
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