Problem 10 - Elementary Algebra

Question

Eight more than a quantity is \(37 .\) What is the original quantity?

Step-by-Step Solution

Answer

Answer: The original quantity is 29.

Step 1: Write down the equation

Write down the equation as x + 8 = 37, where x is the original quantity.

Step 2: rearrange the equation

To solve for x, we will separate x from the constant (8) on one side of the equation. We can do this by subtracting 8 from both sides: x + 8 - 8 = 37 - 8.

Step 3: Subtract and simplify

Perform the subtraction: x = 37 - 8. Simplify to get x = 29. The original quantity is 29.

Key Concepts

Algebraic Expressions

Imagine algebraic expressions as a special language to summarize math problems. They are made up of numbers, variables (letters that represent unknown values), and operation symbols like plus (+) and minus (-). Think of it like a recipe that tells you how to combine different ingredients. For example, 'eight more than a quantity' can be expressed as the expression \( x + 8 \), where \( x \) is the quantity we want to discover, much like a mystery ingredient.

In the algebraic expression \( x + 8 \), \( x \) stands in for the unknown quantity, and the number 8 is being added to it. The expression sets up information that we can manipulate using algebraic rules to find out what \( x \) is. It's like having a box with a hidden treasure inside, and algebra gives us the key to unlock the box and find the treasure (the value of \( x \) in our case).

Equations with Variables

An equation with variables serves as a balance scale. Just as a scale is in balance when the weights on both sides are equal, an equation is in balance when the expressions on both sides are equal. In our case, we're working with the equation \( x + 8 = 37 \), where \( x \) is our variable.

This equation tells us that when 8 is added to our mystery number \( x \), it will equal 37. It's a bit like saying, 'When I add 8 apples to the unknown number of apples in this bag, I have 37 apples in total.' Our goal is to figure out how many apples were in the bag to start with, which means finding the value of \( x \).

To maintain the balance of the equation when searching for \( x \) we perform the same operation (like subtraction) on both sides of the equation. This ensures accuracy, just like carefully adding or removing the same weight from both sides of a scale.

Subtraction in Algebra

Subtraction in algebra follows the same basic principles as arithmetic subtraction, but it also includes dealing with variables. When we subtract a number from both sides of an equation, we're acting like detectives, removing the same clue from both sides to get closer to the answer. In our example, \( x + 8 - 8 = 37 - 8 \), we subtract 8 from each side to maintain equality and isolate the variable \( x \).

After subtracting, we simplify the equation to \( x = 29 \). This simplified equation tells us that the original quantity or 'the unknown number of apples' is 29. Like peeling layers off an onion, subtraction helps us strip away the extra pieces until we're left with what we're looking for: the value of the variable.