Problem 37 - Prealgebra

Question

Complete the following tables. $$\begin{array}{|c|c|} \hline \text { First } & \text { Second } & \text { Their } \\ \text { Number } & \text { Number } & \text { Sum } \\ x & y & x+y \\ \hline-5 & -3 & \\ \hline-5 & -4 & \\ \hline-5 & -5 & \\ \hline-5 & -6 & \\ \hline-5 & -7 & \\ \hline \end{array}$$

Step-by-Step Solution

Answer

Sums: -8, -9, -10, -11, -12.

Step 1: Understand the Problem

We have a table with pairs of numbers and we need to find their sums. The first number in all the pairs is \(-5\) and the second number varies from \(-3\) to \(-7\). We need to fill in the column for \(x + y\), which represents the sum of the first and the second number.

Step 2: Calculate the First Sum

For the first pair, the numbers are \(-5\) and \(-3\). Their sum is calculated as follows:\[ x + y = -5 + (-3) = -8 \]So, the sum for this pair is \(-8\).

Step 3: Calculate the Second Sum

For the second pair, the numbers are \(-5\) and \(-4\). Calculate their sum:\[ x + y = -5 + (-4) = -9 \]Thus, the sum for this pair is \(-9\).

Step 4: Calculate the Third Sum

For the third pair, the numbers are \(-5\) and \(-5\). Their sum is:\[ x + y = -5 + (-5) = -10 \]Thus, the sum for this pair is \(-10\).

Step 5: Calculate the Fourth Sum

For the fourth pair, the numbers are \(-5\) and \(-6\). Calculate their sum:\[ x + y = -5 + (-6) = -11 \]So, the sum is \(-11\).

Step 6: Calculate the Final Sum

For the last pair, the numbers are \(-5\) and \(-7\). Their sum is:\[ x + y = -5 + (-7) = -12 \]Hence, the sum for this pair is \(-12\).

Step 7: Fill in the Table

We can now fill in the 'Their Sum' column in the table with the sums we calculated:- For \(-5\) and \(-3\), the sum is \(-8\).- For \(-5\) and \(-4\), the sum is \(-9\).- For \(-5\) and \(-5\), the sum is \(-10\).- For \(-5\) and \(-6\), the sum is \(-11\).- For \(-5\) and \(-7\), the sum is \(-12\).

Key Concepts

Negative Numbers

When working with integer addition, especially negative numbers, one must be cautious about direction on the number line. A negative number points left, and when you add a negative number, you move further left. This makes the sum of two negative numbers more negative.
For example, if you think of

• adding \(-5\) to \(-3\), consider that both numerals are negative.
• Their sum \(-5 + (-3) = -8\).

It's like saying you found yourself even deeper in debt. Recognizing how negative numbers behave in addition is vital for filling tables and equations correctly.

Table Completion

In math exercises, completing tables serves to structure your calculations and help spot patterns. With integer addition, specifically using one fixed number like \(-5\) as provided in the problem, filling the table becomes systematic. Begin by calculating each cell based on the given instructions.
Here’s a brief breakdown:

• Column one: unchanged number (\(-5\) remains constant.)
• Column two: varies across the rows, from \(-3\) to \(-7\).
• Sum column: derived value by adding the fixed and varying numbers.

By completing each row step-by-step, each sum gives direct feedback into how negative summation works. It’s an organized mode to actively engage with basic arithmetic tasks.

Basic Arithmetic Operations

Understanding basic arithmetic operations, especially in integer addition, lays a fundamental math groundwork. When adding integers:

• Positive plus positive equals more positive results.
• Negative plus negative equals a more negative result.
• Mixed (positive and negative) results depend on which has the greater absolute value, determining the sign of the result.

Here, our exercise primarily dealt with negative numbers and showcased that summing two negatives pushes further left on the number line. Ensuring clarity in addition helps perform accurate calculations without mistakes. This concept is crucial not only for table completions but any numerical problem-solving task.