Problem 14 - Intermediate Algebra

Question

What number should be a. subtracted from 5 to obtain \(0 ?\) b. added to 5 to obtain \(0 ?\)

Step-by-Step Solution

Answer

Subtract 5 from 5 to get 0; add -5 to 5 to get 0.

Step 1: Understanding Subtraction

To find the number that should be subtracted from 5 to obtain 0, we are looking for the number that when taken away from 5 will result in 0.

Step 2: Set Up the Equation for Subtraction

Let the number be represented by \(x\). The equation can be set up as follows: \[5 - x = 0\]

Step 3: Solve the Subtraction Equation

Rearrange the equation to solve for \(x\):\[5 = x\]Therefore, the number that should be subtracted from 5 to obtain 0 is 5.

Step 4: Understanding Addition

To find the number that should be added to 5 to obtain 0, we are looking for a number that when added to 5 gives a sum of 0.

Step 5: Set Up the Equation for Addition

Let the number be represented by \(y\). The equation can be set up as follows: \[5 + y = 0\]

Step 6: Solve the Addition Equation

Rearrange the equation to solve for \(y\):\[y = -5\]Hence, the number that should be added to 5 to obtain 0 is -5.

Key Concepts

Subtraction

Subtraction is a mathematical operation that involves taking one number away from another. It is often used when you want to find out what remains after something is removed or taken away. When you subtract a number from another, you are essentially finding the difference between the two numbers.
Subtraction can be represented with a subtraction equation, such as:

• \(5 - x = 0\)

Here, \(5\) is the original number, \(x\) is the number we need to subtract, and \(0\) is the desired result.
By rearranging the equation, we can solve for \(x\). If \(5 - x = 0\), then \(x\) must equal 5, because \(5 - 5 = 0\). This calculation shows that when you remove 5 from 5, nothing is left, achieving the desired result of 0.

Addition

Addition is the basic process of combining two or more numbers to find their total. It's the opposite of subtraction, as instead of taking away, you are adding two numbers together. In this case, we aim to find which number can be added to another to produce a specified sum.
In the equation:

• \(5 + y = 0\)

\(5\) is the original number, \(y\) is the unknown number to be added, and \(0\) is the desired result.
To solve for \(y\), you need to rearrange the equation, leading to the conclusion that \(y = -5\), because \(5 + (-5) = 0\). This tells us that adding \(-5\) effectively cancels out 5, reaching a total of 0. This shows how addition with negative numbers can result in a lessened value.

Integer Solutions

In mathematics, integers are whole numbers that can be positive, negative, or zero. Integer solutions are solutions to equations that are whole numbers, without fractions or decimals.
When solving equations such as these, you find integer solutions by determining what whole numbers satisfy the equation. For instance:

• For the subtraction equation: \(5 - x = 0\), \(x\) must be 5, which is a whole number.
• For the addition equation: \(5 + y = 0\), \(y\) must be \(-5\), which is also a whole number.

These integer solutions confirm that solving simple arithmetic equations often leads back to whole numbers, providing clear, understandable outcomes. When working with basic equations, it's important to check that your answers are indeed integers when the problem requires it, ensuring they make sense in the context of the problem.