Problem 37 - Prealgebra

Question

Evaluate each expression if \(x=9, y=4,\) and \(z=12\) $$(5 z-4 x)+3 y$$

Step-by-Step Solution

Answer

The value of the expression is 36.

Step 1: Substitute Variables

Substitute the given values for \(x\), \(y\), and \(z\) into the expression \((5z - 4x) + 3y\). With \(x=9\), \(y=4\), and \(z=12\), the expression becomes \( (5(12) - 4(9)) + 3(4) \).

Step 2: Calculate Inside the Parentheses

Calculate \(5(12)\) and \(4(9)\). This gives \(60\) for \(5(12)\) and \(36\) for \(4(9)\). Substitute these into the expression: \((60 - 36) + 3(4)\).

Step 3: Simplify Inside the Parentheses

Subtract \(36\) from \(60\) to simplify inside the parentheses: \(60 - 36 = 24\). The expression is now \(24 + 3(4)\).

Step 4: Calculate Remaining Multiplication

Multiply \(3\) by \(4\) to continue simplifying: \(3(4) = 12\). Substitute this into the expression: \(24 + 12\).

Step 5: Add the Results

Add \(24\) and \(12\) together: \(24 + 12 = 36\). Thus, the value of the expression is \(36\).

Key Concepts

Substitute Variables

Substituting variables is like using a key to unlock a math problem. When you're given specific numbers for variables, you replace each variable in the expression with its given value. This transforms the expression from a general statement into a specific, solvable equation.

In our example, the expression \[(5z - 4x) + 3y\] becomes much clearer when we replace \(z\) with 12, \(x\) with 9, and \(y\) with 4. This substitution results in the expression: \[(5(12) - 4(9)) + 3(4)\]. This new expression is much easier to handle because it’s now all numbers, ready for further calculations.

Evaluate Expressions

When we evaluate an expression, we're essentially doing the math to find out what number the expression equals. This is often what you're tasked with when given a specific problem.

After substituting the variable values, the next step is to "do the math." Start by handling operations inside any parentheses, like \(5(12) - 4(9)\), making sure to multiply before you subtract. This results in \(60 - 36\). Evaluating this further reduces it to \(24\), turning the expression into \[(24) + 3(4)\]. Now it’s ready for the next step, which is simplifying further to find the final answer.

Simplify Expressions

Simplifying an expression means making it as straightforward as possible. This involves reducing complex parts of the expression to single numbers.

For our problem, after calculating \(5(12) = 60\) and \(4(9) = 36\), we simplified the parentheses: \(60 - 36 = 24\). The expression was then simplified to: \[24 + 3(4)\].

Simplification ensures that the expression is easy to compute. It’s important to handle one part of the equation at a time, making sure each step leads us closer to finding the value of the whole expression.

Order of Operations

The order of operations is like having a set of rules that tells us the order in which calculations should be completed to solve an expression correctly. You're likely familiar with PEMDAS/BODMAS, standing for Parentheses/Brackets, Exponents/Orders (powers and roots, etc.), Multiplication, Division, Addition, and Subtraction.

In the given exercise, we first managed the operations inside the parentheses, handling multiplication, and subtraction. Then, we moved on to multiplication outside the parentheses: \(3(4) = 12\). Finally, we performed the addition operation, which combines all the simplified parts together \(24 + 12\) to give us the final result of \(36\).

Following the order of operations ensures that everyone calculates expressions in the same way and gets the correct answers consistently.