Problem 38 - Elementary Algebra

Question

Translate each sentence to a mathematical statement and then simplify. Subtract -5 from the sum of 10 and -3 .

Step-by-Step Solution

Answer

The simplified mathematical statement is 12.

Step 1: Translate the words into a mathematical expression

The sentence states 'Subtract -5 from the sum of 10 and -3'. First, we find the sum of 10 and -3, which is written in mathematical terms as \( 10 + (-3) \). Then, subtract -5 from this sum, which gives the expression \( (10 + (-3)) - (-5) \).

Step 2: Simplify the expression inside the parentheses

Calculate the sum inside the parentheses: \( 10 + (-3) \). This is equivalent to subtracting 3 from 10, resulting in \( 7 \). So, now the expression becomes \( 7 - (-5) \).

Step 3: Simplify the subtraction of negative numbers

Subtracting a negative number is equivalent to adding its positive counterpart. Therefore, \( 7 - (-5) \) is the same as \( 7 + 5 \).

Step 4: Complete the addition

Perform the addition: \( 7 + 5 = 12 \).

Key Concepts

Simplifying Expressions

Simplifying expressions involves reducing a mathematical statement to its simplest form. Let's break it down with our example:

• The original mathematical expression we formed was \((10 + (-3)) - (-5)\).
• First, take care of any operations inside parentheses. In this case, \((10 + (-3))\) becomes easier to handle first.

To simplify the expression within the parentheses, consider what \(10 + (-3)\) means:

• \(10\) is being combined with a negative \(3\).
• This is the same as subtracting \(3\) from \(10\), which results in \(7\).

Now, we have the expression \(7 - (-5)\). Our next step in simplification is understanding how to deal with the subtraction of a negative number.

Adding Negative Numbers

Adding negative numbers can initially seem confusing, but it follows straightforward rules. In our example, we dealt with \(10 + (-3)\).

• When adding a negative number, envision moving backward by that number's value.
• Thus, adding \(-3\) to \(10\) means you move backward three steps on the number line starting at \(10\). This gets you to \(7\).

In essence, adding a negative number is similar to subtraction. You're effectively reducing the initial positive number by the value of the negative number included in the expression.

Subtraction of Negative Numbers

Subtracting negative numbers can often trip up learners, as it involves an extra step in understanding traditional subtraction. For this topic, consider the expression \(7 - (-5)\).

• Think of subtraction of a negative number like taking away a debt, or flipping the sign.
• When you encounter \(7 - (-5)\), it's the same as turning the \(-5\) into a positive and adding it, so \(7 + 5\).

This flip occurs because subtracting a negative is akin to adding a positive. After making this adjustment, the math becomes easy, leading to your final result of \(7 + 5 = 12\). Always remember, double negatives transform into a positive when it comes to subtraction in mathematics.