Problem 39 - Prealgebra

Question

Use the rule for order of operations along with the rules for addition, subtraction, and multiplication to simplify each of the following expressions. $$2(-5)+3(-4)$$

Step-by-Step Solution

Answer

The simplified expression is \(-22\).

Step 1: Apply the order of operations

In the expression \(2(-5) + 3(-4)\), we need to follow the order of operations (PEMDAS/BODMAS). In this case, apply multiplication first, as there are no parentheses to resolve further.

Step 2: Resolve the multiplications

Multiply the coefficients in the expression: \(2(-5) = -10\) and \(3(-4) = -12\). The expression now becomes \(-10 + (-12)\).

Step 3: Simplify the expression

Sum up the results of the multiplications: \(-10 + (-12) = -22\). This combines the two negative results to give a single number.

Key Concepts

Understanding Multiplication in Expressions

When you encounter a mathematical expression, multiplication is typically the first operation to handle after dealing with any parentheses. In the context of equations like \(2(-5) + 3(-4)\), multiplication should be resolved before moving on to addition or subtraction. Here’s how it works:- Identify terms that involve multiplication. Here we focus on \(2(-5)\) and \(3(-4)\).- Perform the multiplication steps: - Multiply \(2\) by \(-5\) to get \(-10\). - Multiply \(3\) by \(-4\) to get \(-12\).Always remember: Multiplying two numbers, where one or both are negative, requires you to pay attention to the signs:- Positive times negative equals negative.- Negative times negative equals positive.By handling these multiplication steps first, you're setting a foundation to simplify and resolve the entire expression correctly.

Addition and Subtraction Rules

Once you have completed the multiplication in your expression, the next step involves addressing addition and subtraction. In our example, after multiplying, we're left with \(-10 + (-12)\).By following these steps, you'll accurately simplify the expression:- Overlook the traditional idea of 'subtract' and instead think of adding negative numbers.- Calculate \(-10 + (-12)\) by recognizing that adding a negative number is equivalent to subtracting. So, you add their absolute values: - Find the sum of \(10\) and \(12\) to get \(22\).- Since both numbers are negative, add their absolute values and keep the negative sign: - Resulting value is \(-22\).This logic holds:- Adding negatives strengthens the negative value.- Subtracting a negative is equivalent to adding a positive.

Expression Simplification Techniques

Simplification of expressions such as \(2(-5) + 3(-4)\) requires a systematic approach involving the order of operations known as PEMDAS/BODMAS. Let’s break it down:- **Parentheses first:** Always check for nested operations inside parentheses and resolve them. In this case, no further reduction is necessary.- **Exponents next:** If there were any, you'd handle them before multiplication/division.- **Multiplication and Division:** As discussed, handle from left to right.- **Addition and Subtraction:** Approach these last, also from left to right.After all calculations, check your final answer for:- Correctness concerning order and mathematical rules.- Simplification, ensuring no further reduction.In our example, following these steps without deviation results in the accurate expression simplification to \(-22\). This step-by-step method ensures precision and clarity.