Problem 10 - Algebra A Combined Function

Question

Solve each equation. Check each solution. See Examples 1 through \(6 .\) \(\frac{1}{8} v=\frac{1}{4}\)

Step-by-Step Solution

Answer

The solution is \( v = 2 \).

Step 1: Understand and Set Up the Equation

The given equation is \( \frac{1}{8}v = \frac{1}{4} \). This is a simple linear equation where we want to solve for \( v \). Our goal is to isolate \( v \) on one side of the equation.

Step 2: Multiply to Isolate the Variable

To solve for \( v \), we need to get rid of the fraction \( \frac{1}{8} \). We do this by multiplying both sides of the equation by the reciprocal of \( \frac{1}{8} \), which is \( 8 \). This gives us \( v = 8 \times \frac{1}{4} \).

Step 3: Perform the Multiplication

Calculate \( 8 \times \frac{1}{4} \). To perform the multiplication, multiply numerators and denominators: \( 8 \times \frac{1}{4} = \frac{8 \times 1}{1 \times 4} = \frac{8}{4} \).

Step 4: Simplify the Result

Simplify \( \frac{8}{4} \) by dividing both the numerator and the denominator by 4: \( \frac{8}{4} = 2 \). Thus, \( v = 2 \).

Step 5: Check the Solution

Substitute \( v = 2 \) back into the original equation to verify the solution: \( \frac{1}{8} \times 2 = \frac{2}{8} = \frac{1}{4} \). Since the left side equals the right side of the original equation, the solution is correct.

Key Concepts

Reciprocal Multiplication

When solving linear equations, one effective method to isolate a variable, such as in our equation \( \frac{1}{8}v = \frac{1}{4} \), is using reciprocal multiplication. The reciprocal of a fraction is simply inverting its numerator and denominator. For instance, the reciprocal of \( \frac{1}{8} \) is \( 8 \) because it can be written as \( \frac{8}{1} \). This technique is powerful because multiplying any number by its reciprocal results in \( 1 \). Therefore, when we multiply both sides of the equation by \( 8 \), \( \frac{1}{8} \times 8 = 1 \), effectively isolating \( v \) on the left side of the equation. Applying reciprocal multiplication helps simplify equations and allows us to solve them in a few straightforward steps. Always remember, the main goal is to make one side of the equation just the variable, which we can achieve by removing fractions using their reciprocals.

Fraction Simplification

After using reciprocal multiplication, we often end up with fractions that need simplification. For the equation \( v = \frac{8}{4} \), simplifying is essential to finding the most reduced form of the solution. Fraction simplification involves reducing a fraction to its simplest form. This is done by dividing both the numerator and the denominator by their greatest common divisor (GCD). In our case, \( \frac{8}{4} \) can be simplified by dividing both 8 and 4 by 4, their GCD, resulting in \( 2 \). Simplifying fractions not only makes them easier to interpret, but it also ensures accuracy in mathematical communication. Always double-check your simplifications to confirm that you have reached the simplest form.

Substitution Method

Once you've solved for the variable, it's crucial to verify your solution, and substitution is a perfect method for this task. In the original equation \( \frac{1}{8}v = \frac{1}{4} \), substituting our found value of \( v = 2 \) back into the equation helps confirm our solution is correct. Start by replacing \( v \) with 2: \( \frac{1}{8} \times 2 \). Perform this multiplication to see if both sides of the equation remain equal, i.e., \( \frac{2}{8} = \frac{1}{4} \). If both sides are equal after substitution, then your solution is verified. This method helps ensure accuracy and builds confidence in solving linear equations.

Equation Verification

Equation verification is the final step in solving any equation. It confirms that your solution satisfies the original problem. After simplifying and substituting back into the equation, check whether the left and right sides balance. For \( \frac{1}{8}v = \frac{1}{4} \), after solving, we substituted \( v = 2 \) back and got \( \frac{2}{8} = \frac{1}{4} \). Since this statement is true, it confirms that our solution is correct. Verification acts as a checkpoint ensuring that we've followed the right steps and reached the correct conclusion. Always take an extra moment to verify your solutions - it's a step that saves time and effort in identifying mistakes early in mathematical problem-solving.