Problem 51 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

The value of the expression is \(-\frac{5}{3}\).

Step 1: Understand the Expression

The expression given is \( \frac{x^{2}-y+2z}{3x} \). We need to substitute the values of \( x = -2 \), \( y = 0 \), and \( z = 3 \) into this expression and simplify.

Step 2: Substitute the Values

Replace \( x \) with \(-2\), \( y \) with \(0\), and \( z \) with \(3\) in the expression \( \frac{x^{2}-y+2z}{3x} \). This gives us:\[\frac{(-2)^{2} - 0 + 2(3)}{3(-2)}\]

Step 3: Simplify the Numerator

Calculate the expression in the numerator: 1. Find \((-2)^{2}\) which is \(4\).2. Calculate \(2 \times 3\) which is \(6\).3. Combine these results to get \(4 + 6 - 0 = 10\).

Step 4: Simplify the Denominator

Calculate the denominator, which is \(3(-2)\):\(3 \times (-2) = -6\).

Step 5: Divide the Numerator by the Denominator

Divide the simplified numerator by the simplified denominator:\[\frac{10}{-6} = -\frac{5}{3}\]

Step 6: Final Result

The simplified result of the expression is \(-\frac{5}{3}\).

Key Concepts

Substitution Method

The substitution method is a fundamental technique in algebra used to simplify expressions or solve equations by replacing variables with their numerical values. When tackling problems like the one in our exercise, the first step is to identify the variables present in the expression. Here, we have the variables: \(x\), \(y\), and \(z\). Once identified, substitute each variable with the given specific numbers. This transforms the abstract expression into a numerical one that can be simplified further.

For example, in the expression \( \frac{x^{2}-y+2z}{3x} \), we substitute \(x = -2\), \(y = 0\), and \(z = 3\). This changes the expression to:

• \[ \frac{(-2)^{2} - 0 + 2(3)}{3(-2)} \]

Correct substitution is critical since any mistakes can lead to incorrect results. Therefore, it's always good practice to replace the variables and immediately check each part of the expression for accuracy. This confirms that the substitution is performed correctly and that we are set to advance to the next step, which is simplification.

Simplifying Expressions

Simplifying expressions is a crucial algebra skill that involves reducing a complex expression into its simplest form, making it more understandable and easier to work with. After substitution, as seen previously, we often end up with numbers and operations in both the numerator and denominator of a fraction that need simplifying.

It begins with the numerator. We calculate each operation:

• \((-2)^2\) results in \(4\) because any number squared results in a positive number.
• The term \(2 \times 3\) gives us \(6\).
• Add these with the value \(-y\), which is \(0\), to get \(4 + 6 - 0 = 10\).

This simplification of the numerator allows us to work with a single number rather than a series of operations.

Next, we simplify the denominator, \(3 \times (-2)\), to obtain \(-6\).

Finally, we carry out the division \(\frac{10}{-6}\) which simplifies to \(-\frac{5}{3}\) by finding the greatest common divisor (\(2\) in this case) and reducing the fraction accordingly. Simplifying expressions easier makes it easier to interpret and use in subsequent calculations or analyses.

Fraction Operations

Fraction operations are often an area where students encounter difficulties, yet they form an integral part of algebra. Fractions consist of a numerator and a denominator and represent a division or ratio. Successfully simplifying or manipulating expressions involving fractions builds on understanding these basic operations.

In our exercise, after simplifying both parts separately, we had the fraction \(\frac{10}{-6}\). To simplify this fraction, remember:

• Both the numerator and the denominator should be divided by their greatest common factor. Here, that factor is \(2\).
  
• Divide \(10\) by \(2\) to get \(5\).
• Divide \(-6\) by \(2\) to get \(-3\).

This yields the simplified fraction \(-\frac{5}{3}\).

Understanding how to handle negative signs is also crucial. Here, either the numerator or the denominator alone can hold the negative sign, but conventionally, it remains in the numerator: \(-\frac{5}{3}\). Practicing these operations ensures proficiency in manipulating and interpreting fractions, crucial for solving complex algebraic problems.