Problem 39 - Prealgebra

Question

Evaluate each expression if \(x=9, y=4,\) and \(z=12\) $$2 x+(4 z-13)-5$$

Step-by-Step Solution

Answer

48

Step 1: Substitute Variables

To evaluate the expression, first substitute the given values of the variables into the expression. Given that \( x = 9 \), \( y = 4 \) (note \( y \) is not needed here), and \( z = 12 \), substitute these into the expression \( 2x + (4z - 13) - 5 \). This gives us \( 2(9) + (4(12) - 13) - 5 \).

Step 2: Multiply Coefficients

First, handle the parts of the expression with multiplication: calculate \( 2 \times 9 = 18 \) and \( 4 \times 12 = 48 \). This simplifies the expression to \( 18 + (48 - 13) - 5 \).

Step 3: Simplify Parentheses

Next, simplify inside the parentheses: calculate \( 48 - 13 = 35 \). This results in \( 18 + 35 - 5 \).

Step 4: Perform Addition and Subtraction

Finally, perform the addition and subtraction in the sequence they appear: first, add \( 18 + 35 = 53 \), then subtract \( 5 \), which gives \( 53 - 5 = 48 \).

Key Concepts

Substitution

Understanding how to substitute variables is crucial in Algebra. When you see an expression like \(2x + (4z - 13) - 5\), the letters \(x\) and \(z\) are placeholders for numbers. Once you know what number each letter represents, you can precisely calculate the expression's value. Here, we know \(x = 9\) and \(z = 12\). **Substitute each number for its corresponding variable**:

• Replace \(x\) with 9: This transforms \(2x\) into \(2 \times 9\).
• Replace \(z\) with 12: This changes \(4z\) into \(4 \times 12\).

You now deal with numbers alone. This step makes the algebraic expression more like a standard arithmetic problem, where the only task is to calculate the expression's value. Substitution transforms abstract letters into concrete numbers, making your task clearer and more straightforward.

Order of Operations

Once you've performed substitution, you'll need to follow the order of operations to solve the expression correctly. The order of operations helps you handle complex calculations by dictating which steps to take first. In general, remember the acronym **PEMDAS**:

• **P**arentheses: Simplify expressions inside parentheses first.
• **E**xponents: Calculate powers and roots (not applicable here).
• **M**ultiplication and **D**ivision: Perform these operations inside-out from left to right.
• **A**ddition and **S**ubtraction: Solve these from left to right.

For \(2(9) + (4(12) - 13) - 5\), you tackle the parentheses and multiplication early on. First, the multiplication:

• Calculate \(2 \times 9 = 18\).
• Calculate \(4 \times 12 = 48\).

Next, address the expression within the parentheses:

• Simplify: \(48 - 13 = 35\).

This structure ensures you gradually advance towards the final solution without errors.

Evaluating Expressions

After substitution and following the order of operations, you reach the last step: evaluating the expression. This involves executing calculations step by step to find the final answer. With the expression simplified to \(18 + 35 - 5\), use straightforward arithmetic:

• Add: \(18 + 35 = 53\).
• Subtract: \(53 - 5 = 48\).

Evaluating expressions combines all previous concepts. You first turned variables into numbers via substitution, then calculated accurately through the order of operations, leading to an exact result.By practicing these steps, you'll improve your ability to handle complex algebraic tasks, eventually doing them reliably and quickly. Mastery of evaluation enables you to interpret expressions correctly, providing valuable skills for tackling more advanced mathematics.