Problem 10 - Prealgebra

Question

Solve each equation. Check your solution. $$\frac{h}{4}=6$$

Step-by-Step Solution

Answer

The solution to the equation is \( h = 24 \).

Step 1: Multiply to Isolate the Variable

To solve for \( h \) in the equation \( \frac{h}{4} = 6 \), multiply both sides by 4 to remove the fraction. This gives us:\[h = 6 \times 4\] which simplifies to:\[h = 24\]

Step 2: Verify the Solution

Substitute \( h = 24 \) back into the original equation to check the solution:\[\frac{24}{4} = 6\] Simplify the left side:\[6 = 6\] Since both sides are equal, our solution \( h = 24 \) is correct.

Key Concepts

Checking Solutions

After solving an equation, it's critical to verify your solution to confirm that it works in the original equation. This step isn't just a good habit; it's a necessary part of solving equations. To check your solution:

• Take the value you found for the variable and substitute it back into the original equation.
• Simplify both sides of the equation using standard arithmetic operations to see if they equal.

If both sides of the equation balance out, this indicates that the solution is correct. In the exercise where you solved for \(h\) and found \( h = 24 \), substituting it back into \( \frac{h}{4} = 6 \) results in \( 6 = 6 \), confirming accuracy. Making a habit of checking your solutions improves your confidence and ensures that you have not made any calculation mistakes. This practice may seem redundant, but it's crucial for accuracy, especially with more complex equations.

Multiplying to Isolate Variables

Solving equations often requires isolating the variable, which means having it alone on one side of the equation. A common tool for doing this involves multiplication, especially when dealing with fractions.Here's how you use multiplication to isolate a variable:

• Identify terms dividing the variable, such as fractions or coefficients.
• Multiply both sides of the equation by the same number to cancel out these terms. This maintains equality.

For our exercise, \( \frac{h}{4} = 6 \), you multiplied both sides by 4 to eliminate the fraction, yielding \( h = 24 \). Using multiplication in this way allows you to transform equations into a simpler form where the solution becomes apparent. Consistency in performing the same operation on both sides of an equation is key to maintaining the balance needed for a correct solution.

Fractions in Equations

Fractions can sometimes make solving equations appear tricky. However, with the right approach, they can be easily managed.When you encounter fractions in an equation:

• Consider multiplying both sides of the equation by the denominator to eliminate the fraction. This simplifies the equation quickly.
• Ensure that all fractions are handled before proceeding to solve the equation fully.

In the exercise \( \frac{h}{4} = 6 \), the fraction \( \frac{h}{4} \) can be thought of as dividing \( h \) by 4. By multiplying everything by 4, the equation simplifies without fractions to examine, resulting in \( h = 24 \). Understanding how to manipulate fractions within an equation is key to mastering algebra and makes equations more straightforward to solve. By removing fractions early, you reduce the chance of errors and make subsequent operations cleaner and clearer.