Problem 56 - Algebra 1

Question

Evaluate the expression for the given value(s) of the variable(s). $$\frac{28-4 x}{y} \text { when } x=2 \text { and } y=\frac{1}{2}$$

Step-by-Step Solution

Answer

The evaluated expression equals 40.

Step 1: Substitute the value of \(x\) into the equation

Replace \(x\) with 2 in the equation to get \(\frac{28-4*2}{y}\) which simplifies to \(\frac{28-8}{y}\).

Step 2: Simplify the numerator

The subtraction in the numerator gives \(20/y\).

Step 3: Substitute the value of \(y\) into the equation

Now replace \(y\) with 0.5 in the equation to get \(\frac{20}{0.5}\).

Step 4: Calculate the division

The division \(\frac{20}{0.5}\) equals 40.

Key Concepts

Substitution Method

The substitution method is a vital technique in mathematics used when you are given a specific value for a variable. This method lets you replace a variable with its given number.
In this exercise, you have the expression \( \frac{28 - 4x}{y} \). We need the value of this expression when \(x = 2\) and \(y = \frac{1}{2}\). The starting point is

• Identify the variables and their corresponding values.
• Substitute these values into the expression.

This transforms the expression so that you can then perform arithmetic operations, like subtraction or division, with just numbers.

Division

Division, a basic arithmetic operation, involves finding out how many times one number is contained within another. It's crucial to understand the expression \(\frac{a}{b}\) as asking, "How many \(b's\) fit into \(a\)?"
In our given expression \(\frac{20}{0.5}\):

• The numerator is 20, representing how much or how many of something you have.
• The denominator, 0.5, indicates the size or amount you are using to divide the numerator.

Calculating this division gives you the result:
\(20 \div 0.5 = 40\).
Remember, dividing by a fraction is like multiplying by its reciprocal, so \(0.5\) being half makes the division seem larger.

Simplifying Expressions

Simplifying expressions means reducing them to their simplest form. This involves performing all possible operations to make it easier to understand and calculate.
For example, in the expression \(28 - 4x\), you simplify by calculating \(28 - 4 \times 2\):

• Multiply: \(4 \times 2 = 8\).
• Subtract: \(28 - 8 = 20\).

This process removes all potential ambiguities in your expression by reducing it to a single number or a basic operation.

Numerical Substitution

Numerical substitution is closely related to the substitution method. This involves replacing variables with specific numerical values and performing arithmetic calculations.
In our exercise:

• Start with the expression \(\frac{28 - 4x}{y}\).
• First, substitute \(x = 2\) to get \(\frac{28 - 8}{y}\).
• Then, substitute \(y = \frac{1}{2}\) in the expression \(\frac{20}{y}\).

With numerical substitution, calculations become more straightforward, allowing you to solve complex expressions step-by-step using basic arithmetic.