Problem 22 - Elementary Algebra

Question

Evaluate. \(23 x-12,\) where \(x=-14\)

Step-by-Step Solution

Answer

The evaluated result is -334.

Step 1: Understand the Expression

The expression to evaluate is \(23x - 12\), where \(x = -14\). This means wherever there is an \(x\) in the expression, we will substitute it with \(-14\).

Step 2: Substitute the Value of x

Replace the variable \(x\) with \(-14\) in the expression. This yields \(23(-14) - 12\).

Step 3: Multiply

Multiply \(23\) by \(-14\) to get the value. Calculate \(23 \times -14 = -322\).

Step 4: Perform Subtraction

Subtract \(12\) from \(-322\). This gives \(-322 - 12 = -334\).

Step 5: Conclusion: Evaluate the Expression

The evaluated result of the expression \(23x - 12\) with \(x = -14\) is \(-334\).

Key Concepts

Substitution Method

The substitution method is a fundamental concept in algebra that helps us to analyze and solve expressions by assigning specific values to variables. In this approach, you replace the variable in an equation or expression with a given numerical value to find the simplified result. For the expression we are working with, which is \(23x - 12\), the variable \(x\) is given to be \(-14\).

• Step 1: Identify the variable \(x\) in the expression.
• Step 2: Replace each occurrence of \(x\) with the number \(-14\).

This substitution transforms the original expression into \(23(-14) - 12\), allowing us to evaluate it further. If you keep the process organized, the substitution method paves the way for reducing complex algebraic expressions to solvable arithmetic tasks.

Multiplication of Integers

Multiplying integers extends the basic rules of arithmetic while also introducing the concept of negative numbers. For our exercise, we needed to multiply \(23\) by \(-14\). Here's how integer multiplication works:

• Positive times positive equals a positive: \( (+) \times (+) = (+) \)
• Negative times negative equals a positive: \( (-) \times (-) = (+) \)
• Positive times negative equals a negative: \( (+) \times (-) = (-) \)

In our case, \(23\) is positive and \(-14\) is negative. Thus, multiplying these yields a negative result: \(23 \times -14 = -322\). Understanding these sign rules is crucial for handling integers, helping you accurately solve expressions involving different signs.

Evaluating Expressions

Evaluating expressions is the process of simplifying them by performing the operations indicated and coming up with a single value. Once the substitution of values and multiplication are complete, you're left with a simpler arithmetic operation. Continuing from the last step, we had the expression \(-322 - 12\).

• First, after the multiplication of integers, you perform the basic arithmetic operation, which in this case, is subtraction.
• Subtract \(12\) from \(-322\). Since both numbers are negative, you add their absolute values, leading to a more negative result: \(-322 - 12 = -334\).

The evaluated result of the expression \(23x - 12\) when \(x = -14\) is \(-334\). Evaluating expressions like this walks you through the critical thinking and computation necessary to solve algebraic problems effectively.