Solving For H: -2/3 * H = -22 - A Math Guide

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Hey guys! Today, we're diving into a common type of math problem: solving for a variable in an equation. Specifically, we're going to tackle the equation -2/3 * h = -22. This might look a little intimidating at first, but trust me, it's totally manageable. We'll break it down step-by-step so you can confidently solve similar problems in the future. So, grab your pencils and let's get started!

Understanding the Equation

Before we jump into solving, let's make sure we understand what the equation is telling us. The equation -2/3 * h = -22 is saying that negative two-thirds times some number, which we're calling 'h', equals negative 22. Our goal is to find the value of 'h' that makes this statement true. In essence, we need to isolate 'h' on one side of the equation. This involves using inverse operations, which are operations that undo each other. Think of it like this: if the equation were saying "something plus 5 equals 10", you'd subtract 5 from both sides to get 'something' by itself. We'll be using a similar approach here, but with fractions and multiplication. The key concept to remember is that whatever operation you perform on one side of the equation, you must also perform on the other side to maintain the balance. This ensures that the equality remains valid. Understanding this fundamental principle is crucial for solving not just this equation, but any algebraic equation you encounter. By keeping the equation balanced, we ensure that the solution we find is accurate and reliable. So, let's move on to the actual steps involved in solving for 'h'!

Step-by-Step Solution

Okay, let's get down to business! Here’s how we can solve the equation -2/3 * h = -22:

1. Isolate 'h' by Multiplying by the Reciprocal

The most common method to isolate 'h' when it's multiplied by a fraction is to multiply both sides of the equation by the reciprocal of that fraction. The reciprocal of a fraction is simply flipping the numerator and the denominator. In our case, the fraction multiplying 'h' is -2/3. So, the reciprocal of -2/3 is -3/2. Remember, the reciprocal maintains the sign, so a negative fraction has a negative reciprocal.

Now, let’s multiply both sides of the equation by -3/2:

(-3/2) * (-2/3 * h) = (-3/2) * (-22)

2. Simplify the Left Side

On the left side, we have (-3/2) multiplied by (-2/3 * h). When you multiply a fraction by its reciprocal, the result is always 1. This is because the numerators and denominators cancel each other out. So, (-3/2) * (-2/3) equals 1. This simplifies our equation to:

1 * h = (-3/2) * (-22)

h = (-3/2) * (-22)

Now, 'h' is isolated on the left side, which is exactly what we wanted!

3. Simplify the Right Side

Now we need to simplify the right side of the equation: (-3/2) * (-22). To multiply a fraction by a whole number, you can think of the whole number as a fraction with a denominator of 1. So, -22 is the same as -22/1. Now we can multiply the fractions:

h = (-3/2) * (-22/1)

To multiply fractions, you multiply the numerators together and the denominators together:

h = (-3 * -22) / (2 * 1)

h = 66 / 2

4. Divide to Find the Value of h

Finally, we divide 66 by 2 to get the value of 'h':

h = 33

So, we've found that h = 33 is the solution to our equation!

Verification of the Solution

It's always a good idea to check your answer to make sure it's correct. To do this, we substitute our solution, h = 33, back into the original equation:

-2/3 * h = -22

-2/3 * 33 = -22

To multiply -2/3 by 33, we can think of 33 as 33/1:

(-2/3) * (33/1) = -22

Multiply the numerators and the denominators:

(-2 * 33) / (3 * 1) = -22

-66 / 3 = -22

Now, divide -66 by 3:

-22 = -22

Since the left side equals the right side, our solution is correct! h = 33 is indeed the value that makes the equation true.

Alternative Method: Clearing the Fraction First

There's another way to solve this equation that some people find easier. Instead of multiplying by the reciprocal right away, we can first clear the fraction from the equation. This involves multiplying both sides of the equation by the denominator of the fraction. In our case, the denominator is 3. Let's see how this works.

1. Multiply Both Sides by the Denominator

Our equation is -2/3 * h = -22. To clear the fraction, we multiply both sides by 3:

3 * (-2/3 * h) = 3 * (-22)

2. Simplify

On the left side, the 3 in front cancels out the 3 in the denominator of the fraction:

-2 * h = 3 * (-22)

On the right side, we multiply 3 by -22:

-2h = -66

3. Divide by the Coefficient

Now we have a simpler equation: -2h = -66. To isolate 'h', we divide both sides by the coefficient, which is the number multiplying 'h'. In this case, the coefficient is -2:

-2h / -2 = -66 / -2

h = 33

We get the same answer, h = 33! This method can be especially useful when dealing with more complex equations involving multiple fractions.

Tips and Tricks for Solving Equations with Fractions

Working with fractions in equations can sometimes be tricky, but here are a few tips and tricks to make the process smoother:

  • Always Check Your Answer: As we demonstrated earlier, plugging your solution back into the original equation is a crucial step. It ensures you haven't made any mistakes along the way.
  • Keep it Balanced: Remember, whatever you do to one side of the equation, you must do to the other. This is the golden rule of equation solving.
  • Simplify First: Before you start solving, look for opportunities to simplify the equation. This might involve combining like terms or reducing fractions.
  • Use the Reciprocal Wisely: Multiplying by the reciprocal is a powerful tool, but make sure you're using the correct reciprocal (flipping the fraction) and that you apply it to both sides.

Real-World Applications

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