Problem 54 - Algebra A Combined Function

Question

If \(x=-2, y=0\), and \(z=3\), find the value of each expression. $$ \frac{4 y-3 x+z}{2 x+y} $$

Step-by-Step Solution

Answer

The expression evaluates to \( -2.25 \).

Step 1: Substitute the given values

Start by substituting the values of the variables into the expression. For this problem, substitute \( x = -2 \), \( y = 0 \), and \( z = 3 \) into the expression \( \frac{4y - 3x + z}{2x + y} \). The expression becomes \( \frac{4(0) - 3(-2) + 3}{2(-2) + 0} \).

Step 2: Simplify the numerator

Work on simplifying the numerator, \( 4(0) - 3(-2) + 3 \). Calculate each part: \( 4 imes 0 = 0 \), \( -3 imes -2 = 6 \), and add the remaining: \( 0 + 6 + 3 = 9 \). So, the numerator simplifies to \( 9 \).

Step 3: Simplify the denominator

Now simplify the denominator, \( 2(-2) + 0 \). Calculate: \( 2 imes -2 = -4 \), so the denominator simplifies to \( -4 \).

Step 4: Calculate the expression

Substitute the simplified numerator and denominator back into the expression: \( \frac{9}{-4} \). Divide 9 by -4 to get \( -2.25 \).

Step 5: Verify the result

Re-evaluate each operation to ensure accuracy: numerator \( 4(0) - 3(-2) + 3 = 9 \) and denominator \( 2(-2) + 0 = -4 \). Calculate \( \frac{9}{-4} \) and verify the result is \( -2.25 \).

Key Concepts

Substitution Method

The substitution method is a simple yet powerful technique used in algebra to replace variables with their given values in an equation or expression. This method helps in solving expressions by turning them into numerical operations. In our example, we are given specific values for the variables: \(x = -2\), \(y = 0\), and \(z = 3\).

We start by substituting these values into the expression \( \frac{4y - 3x + z}{2x + y} \). By doing so, the expression transforms into \( \frac{4(0) - 3(-2) + 3}{2(-2) + 0} \).

The substitution method effectively reduces the number of variables in the equation, allowing us to focus on carrying out arithmetic operations.

When applying this method, it is crucial to accurately substitute the correct value for each variable to avoid calculation errors, which could lead to incorrect results.

Numerator and Denominator Simplification

Simplifying the numerator and denominator separately is a key skill in evaluating algebraic expressions, especially rational expressions.

In the given expression after substitution, we have the numerator as \( 4(0) - 3(-2) + 3 \). Let's break down the steps:

• Calculate \( 4 \times 0 = 0 \)
• Calculate \( -3 \times -2 = 6 \)
• Add the results together: \( 0 + 6 + 3 = 9 \)

Thus, the numerator simplifies to \( 9 \).

For the denominator, after substitution, we have \( 2(-2) + 0 \). This simplifies to:

• Calculate \( 2 \times -2 = -4 \)
• Add \( 0 \), to get \( -4 \)

So, the denominator simplifies to \( -4 \).

Understanding how to simplify each part correctly ensures that you can evaluate the entire expression accurately. Each calculation step might seem small, but accuracy in these details is crucial.

Rational Numbers

Rational numbers are numbers that can be expressed as a fraction or a ratio of two integers, where the denominator is not zero. In mathematical terms, a rational number is written as \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b eq 0 \).

In our exercise, the expression \( \frac{9}{-4} \) is a rational number because it represents a ratio of the integer 9 to the integer -4. After substituting and simplifying the values, we arrive at this fraction.

Rational numbers could be positive, negative, or zero. The arithmetic operation performed here divides the numerator 9 by the denominator -4, resulting in the rational number \(-2.25\), which is a decimal representation.

Understanding rational numbers helps students see the importance of expressions and their simplifications. It aids in grasping how different numbers interact, the significance of division, and how to correctly interpret the results of such calculations.