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Chapter 1: Problem 90

Evaluate each expression for \(x=6, y=-4,\) and \(a=3\) $$ 5 x-4 a^{2} $$

Short Answer

Expert verified
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Step by step solution

01

Replace Variables with Given Values

Identify the given values for the variables: \(x=6\), \(y=-4\), and \(a=3\). Substitute these values into the expression \(5x - 4a^2\).
02

Substitute and Simplify

Substitute \(x=6\) and \(a=3\) into the expression: \[5(6) - 4(3)^2\]. First, calculate the power: \(3^2 = 9\). So the expression becomes \[5(6) - 4(9)\].
03

Perform Multiplication

Calculate the multiplication: \[5(6) = 30\] and \[4(9) = 36\]. Now the expression is: \[30 - 36\].
04

Simplify the Final Result

Subtract the values: \[30 - 36 = -6\]. Therefore, the evaluated expression is \(-6\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Substitution
Substitution is a critical technique in algebra that involves replacing variables with their respective values. Imagine you have an expression, and you want to know its value for certain numbers. This process requires careful replacement of variables with the provided numerical values.
In the given exercise, we have the variables: \( x = 6 \), \( y = -4 \), and \( a = 3 \). The expression provided is \( 5x - 4a^2 \). The first step is to substitute these values into the expression. It transforms the abstract expression into a numerical form:
  • \( x = 6 \)
  • \( a = 3 \)
After substitution, the expression becomes:
  • \( 5(6) - 4(3)^2 \)
This is now ready for the next steps of simplification and calculation.
Simplification
Simplification is the process of reducing complexity in an expression to make calculations easier and clearer. After substitution, the goal is to simplify the expression step by step.
In our example, after substituting the values, we have: \( 5(6) - 4(3)^2 \).
To simplify, follow these steps:
  • First, compute any powers: \( 3^2 = 9 \).
  • Next, substitute this back into the expression: \( 5(6) - 4(9) \).
  • Then, perform the multiplication: \( 5 \times 6 = 30 \) and \( 4 \times 9 = 36 \).
Through these simplification steps, our expression is now: \( 30 - 36 \). Simplification converts our complex initial expression into a form that is straightforward to solve.
Order of Operations
The order of operations is a set of rules that dictates the correct sequence to evaluate a mathematical expression. Remember the acronym PEMDAS:
  • P - Parentheses
  • E - Exponents
  • M/D - Multiplication and Division (from left to right)
  • A/S - Addition and Subtraction (from left to right)
Applying this to our exercise helps ensure accuracy.
Firstly, we deal with exponents and parentheses if present. For our problem, we first solve the exponent: \( 3^2 = 9 \). Then we move on to multiplication: solving \( 5(6) = 30 \) and \( 4(9) = 36 \). Lastly, we handle addition and subtraction: \( 30 - 36 = -6 \).
By following the order of operations, we systematically approach and solve the expression correctly, ensuring every step is accurate and nothing is missed.

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