Problem 42 - Algebra 1: Concepts and Skills

Question

Evaluate the expression for the given value(s) of the variable(s). \(\frac{2-4 x}{y}\) when \(x=2\) and \(y=\frac{1}{2}\)

Step-by-Step Solution

Answer

Therefore, the value of the expression when \(x=2\) and \(y=\frac{1}{2}\) is \(-12\).

Step 1: Substitute the given values

First, substitute the values \(x=2\) and \(y=\frac{1}{2}\) into the expression \( \frac{2 - 4x}{y}\). This results in \( \frac{2 - 4(2)}{\frac{1}{2}}\).

Step 2: Simplify numerator

Perform the multiplication in the numerator to obtain \(2 - 4*2 = 2 - 8\), which simplifies to \(-6\)

Step 3: Final simplification

Now, simplify the entire fraction with the numerator of \(-6\) and denominator of \( \frac{1}{2}\), which results in \(-6 \div \frac{1}{2} = -12\).

Key Concepts

Substitution

Substitution is the first and probably the most straightforward step when evaluating expressions. It's like a simple game of replace-the-variable. In math, the variable acts like a placeholder, similar to an empty box you’re planning to fill in. When you are given specific values for these variables, your job is to "substitute," or replace, each variable with the appropriate number.
In our exercise, we start with the expression \(\frac{2-4 x}{y}\). You are told the values: \(x = 2\) and \(y = \frac{1}{2}\). By substituting these values into the expression, it turns into:

• Replace \(x\) with 2 in \(2-4x\).
• Replace \(y\) with \(\frac{1}{2}\).

Now, the expression is \(\frac{2 - 4(2)}{\frac{1}{2}}\). This substitution step sets the stage for further simplification.

Numerator Simplification

Once variables are substituted, it’s time to turn your attention to simplifying the numerator, which is the top part of a fraction. Simplifying an expression usually involves performing arithmetic operations like addition, subtraction, multiplication, or division.
In this particular expression, \(2 - 4(2)\) is your target. Here’s how you handle it:

• Multiply 4 by 2: \(4 \times 2 = 8\).
• Perform the subtraction: \(2 - 8 = -6\).

This simplification reveals the numerator to be \(-6\). Understanding each step helps you handle more complex expressions and keeps mistakes at bay. Always ensure each step is accurate and straightforward.

Fraction Division

With the numerator simplified, now it's time to simplify the entire fraction through division. When dealing with fractions, division can seem tricky, but it can be made simple with the clear approach.
Here, we have a fraction \(\frac{-6}{\frac{1}{2}}\). When dividing by a fraction, you multiply by its reciprocal. The reciprocal is essentially flipping the numerator and denominator of the fraction you’re dividing by.

• Reciprocal of \(\frac{1}{2}\) is 2, because \(\frac{1}{2}\) flipped is \(2\over 1\), which is simply 2.
• Multiply \(-6\) by 2, which gives us \(-12\).

So, \(-6 \div \frac{1}{2} = -12\). This final step completes the evaluation, providing a satisfying conclusion to any expression management.