Problem 35 - Prealgebra

Question

Evaluate each expression if \(x=9, y=4,\) and \(z=12\) $$7 z-(y+x)$$

Step-by-Step Solution

Answer

The value of the expression is 71.

Step 1: Substitute the Values

Replace each variable in the expression with its given value. In our expression, we replace \(z\) with 12, \(y\) with 4, and \(x\) with 9. The expression becomes \(7 \times 12 - (4 + 9)\).

Step 2: Calculate the Product

First, calculate the product of 7 and \(z\). This is \(7 \times 12 = 84\).

Step 3: Simplify the Parentheses

Calculate the sum inside the parentheses: \(4 + 9 = 13\).

Step 4: Perform the Subtraction

Subtract the result from the parentheses from the product calculated earlier: \(84 - 13 = 71\).

Key Concepts

Variable Substitution

Variable substitution is the process of replacing variables with their given numerical values to solve an expression or an equation. This is a crucial step in prealgebra which helps simplify expressions and make calculations more straightforward. For instance, in the exercise provided, we have variables like \(x\), \(y\), and \(z\) with given values \(x=9\), \(y=4\), and \(z=12\).
To perform variable substitution:

• Identify each variable in the expression.
• Replace each variable with its given numerical value.
• Rewrite the expression using these values.

For example, in the expression \(7z-(y+x)\), replace \(z\) with 12, \(y\) with 4, and \(x\) with 9. The expression then becomes \(7 \times 12 - (4 + 9)\), which now contains only numbers, making it easier to evaluate.

Order of Operations

The order of operations is a critical concept to grasp in prealgebra, ensuring that expressions are simplified correctly. The commonly used acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right)) helps in remembering the sequence in which operations should be performed. Neglecting this order can lead to incorrect results.
In our problem:

• First, handle operations inside parentheses \((4 + 9)\).
• Then, move to multiplication \((7 \times 12)\).
• Finally, conduct the subtraction operation, which is \((84 - 13)\).

By ensuring operations are carried out in this specific order, the expression \(7 \times 12 - (4 + 9)\) results in the correct value of 71. Always remember to work through the brackets first, followed by multiplication, and prioritize these before jumping into addition or subtraction.

Simplifying Expressions

Simplifying expressions involves combining like terms and executing operations using the rules of arithmetic and algebra to reduce an expression to its simplest form. This process makes it easier to interpret and solve mathematical problems, allowing for better accuracy and understanding.
To simplify the given expression \(7 \times 12 - (4 + 9)\):

• First, complete any calculations within parentheses or brackets. Here, \(4 + 9 = 13\).
• Then, perform multiplication: \(7 \times 12=84\).
• Finally, subtract: \(84 - 13\) simplifies directly to 71.

By following these simplification steps, the expression is reduced from its initial format to the more manageable and clear numeric form, 71. This process is especially beneficial for more complex algebraic expressions you may encounter in the future.