Problem 57 - Elementary Algebra

Question

Find the value of each of the following expressions. $$ -4(1-8)+3(10-3) $$

Step-by-Step Solution

Answer

Question: Evaluate the given expression: $$ -4(1-8)+3(10-3) $$ Answer: 49

Step 1: Solve the expressions in parentheses

Evaluate the parenthetical expressions first: $$(1-8) = -7$$ $$(10-3)= 7$$ Now the given expression becomes $$ -4(-7) + 3(7) $$

Step 2: Perform multiplication

Now, we can multiply: $$ (-4)(-7) = 28 $$ $$ (3)(7) = 21 $$ Now, substitute these products back into the expression: $$ 28 + 21$$

Step 3: Perform addition

Finally, we can add these two values: $$ 28 + 21 = 49 $$ So the value of the expression $$ -4(1-8)+3(10-3) $$ is 49.

Key Concepts

Understanding Parentheses in Algebra

Parentheses in algebra are an essential tool, as they tell us which operations to perform first. When you see parentheses, perform the calculations inside them before anything else. This follows the "Order of Operations," often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).

• First, solve any expressions within parentheses.
• Then, proceed with the algebraic operations normally.

In the given problem, we had two sets of parentheses:
- \(1 - 8\) which equals \(-7\), and- \(10 - 3\) which equals \(7\).
By solving these parts first, we transform the complex expression into something more manageable, setting the stage for subsequent operations.

Mastering Multiplication of Negative Numbers

The multiplication of negative numbers can sometimes be confusing, but it's a crucial skill in algebra. The rule is straightforward:

• When you multiply two negative numbers, the result is positive.
• If you multiply a positive with a negative number, the result is negative.

Here's why:
Imagine measuring temperature as below zero (negative) and think of subtracting a daily coldness increment (another negative). It paradoxically becomes warmer (positive).
In our problem:- The calculation \((-4) \times (-7) = 28\) follows the rule of multiplying two negatives, resulting in a positive.
This understanding simplifies the process, reinforcing why practice with such numbers provides confidence in finding solutions.

Simplifying with Addition of Integers

After solving parentheses and multiplying, we often end with addition or subtraction of integers.
Adding integers might involve different scenarios:

• When adding two positive numbers, the result is positive.
• Adding two negative numbers results in a more negative number.
• For a positive and negative number, subtract and keep the sign of the larger absolute value integer.

In the expression, we needed to add the results from our multiplications: \(28 + 21\), both positive.
This direct addition is simple, giving us \(49\). Remember, handling integer addition correctly is key, especially because it often concludes algebraic problems. Applying basic rules consistently unlocks solutions effectively. Keep practicing and these rules will become second nature!