Problem 76 - Fundamentals of Mathematics

Question

For problems \(57-140\), solve each equation. $$ 4 y=\frac{1}{2} $$

Step-by-Step Solution

Answer

The solution is \(y = \frac{1}{8}\).

Step 1: Understand the Equation

We are given the equation \(4y = \frac{1}{2}\). The goal is to find the value of \(y\) that satisfies this equation.

Step 2: Isolate the Variable

To solve for \(y\), we need to isolate \(y\) on one side of the equation. We can do this by dividing both sides of the equation by 4. So the equation becomes: \[ y = \frac{1}{2} \div 4 \]

Step 3: Simplify the Division

Now we perform the division on the right side. Dividing by 4 is the same as multiplying by \(\frac{1}{4}\): \[ y = \frac{1}{2} \times \frac{1}{4} \]

Step 4: Calculate the Result

Multiply the fractions to find \(y\): \[ y = \frac{1}{2} \times \frac{1}{4} = \frac{1}{8} \]

Step 5: Verify the Solution

To ensure the solution is correct, substitute \(y = \frac{1}{8}\) back into the original equation: \[ 4 \times \frac{1}{8} = \frac{4}{8} = \frac{1}{2} \] The left side equals the right side, confirming our solution.

Key Concepts

Solving Equations

Solving equations is a foundational skill in algebra. An equation is a mathematical statement that asserts the equality of two expressions. For example, the equation \(4y = \frac{1}{2}\) asks us to find a value for \(y\) that makes both sides of the equation equal.

To solve an equation, you perform operations to simplify or alter the equation step-by-step until you find the value of the unknown variable. In our example, solving the equation involves isolating \(y\) to find its specific value, which involves manipulating the equation in a balanced way to maintain the equality. This process helps us understand how different mathematical actions affect the equation and provide a solution that satisfies it.

Isolating Variables

Isolating the variable is a critical step in solving equations. It involves rearranging the equation so that the unknown variable, in our case, \(y\), stands alone on one side. This makes it easy to determine its value.

Here’s how you can isolate a variable:

• Perform operations that will "free" the variable from other terms or coefficients. For example, if \(4y = \frac{1}{2}\), you would divide both sides by 4.
• Keep the equation balanced. Whatever operation you apply to one side, apply the same to the other.
• Simplify the equation after isolating the variable. This might mean performing arithmetic on fractions or integers.

Once \(y\) is isolated, you have an expression showing what \(y\) is equal to, allowing you to calculate its value directly.

Fraction Simplification

Fraction simplification is essential when dealing with equations involving fractions. Simplifying a fraction means rewriting it in the simplest form without changing its value.

Here's how we can simplify fractions effectively:

• Look for common factors in the numerator and the denominator and divide them out to simplify the fraction.
• If fractions are involved in division, remember that dividing by a fraction is the same as multiplying by its reciprocal. For example, dividing by \(4\) can be seen as multiplying by \(\frac{1}{4}\).
• Multiply fractions by multiplying tops (numerators) and bottoms (denominators) separately.

Simplifying helps make the numbers more manageable and the equation easier to work with. For instance, \(y = \frac{1}{2} \times \frac{1}{4}\) simplifies to \(y = \frac{1}{8}\), giving a clear, simplified result of our operations.