Linear equations in one variable

Mathematics · Pre-Secondary One Hong Kong Attainment Test · 5 min read

Welcome to the World of Algebra!

Hello there! Are you ready to become a math detective? Today, we are diving into Linear Equations in One Variable. This might sound like a big, scary name, but it is actually like solving a mystery puzzle. In this chapter, you will learn how to find "hidden numbers" and solve real-life problems using simple rules.

Don't worry if this seems tricky at first. Many students feel the same way! We will take it step-by-step, and before you know it, you will be solving for \( x \) like a pro.

1. What is a Variable?

In mathematics, we sometimes don't know a number yet. Instead of leaving a blank space, we use a letter to represent that mystery number. This letter is called a variable.

Common Variables: We usually use letters like \( x \), \( y \), \( a \), or \( n \).
Analogy: Think of a variable as an empty box. We know something is inside the box, and our job is to find out what that number is!

Quick Review:
- Variable: A symbol (usually a letter) that stands for a number.
- Example: In the expression \( x + 5 \), the letter \( x \) is the variable.

2. Expressions vs. Equations

It is very important to know the difference between these two!

Algebraic Expression: A group of numbers and variables (e.g., \( x + 7 \)). It does not have an equal sign.
Equation: A mathematical sentence that says two things are equal (e.g., \( x + 7 = 10 \)). It always has an equal sign \( = \).

Did you know? The word "Equation" comes from the word "Equal." If there is no "equal" sign, it’s not an equation!

Key Takeaway: An equation is like a balance scale. For the scale to stay level, the left side must have the exact same value as the right side.

3. The Golden Rule of Equations

To solve an equation, we want to get the variable all by itself on one side (like \( x = 5 \)). To do this, we use the Golden Rule:

"Whatever you do to one side of the equation, you MUST do to the other side."

If you add 2 to the left side, you must add 2 to the right side to keep the scale balanced.

4. How to Solve Equations (Step-by-Step)

To get the variable alone, we use opposite operations (also called inverse operations). These "undo" what has been done to the variable.

The Opposites:
- The opposite of Plus \( + \) is Minus \( - \)
- The opposite of Minus \( - \) is Plus \( + \)
- The opposite of Multiplication \( \times \) is Division \( \div \)
- The opposite of Division \( \div \) is Multiplication \( \times \)

Example 1: Solving by Subtraction

Solve: \( x + 8 = 15 \)
1. We see \( + 8 \). The opposite is \( - 8 \).
2. Subtract 8 from both sides: \( x + 8 - 8 = 15 - 8 \)
3. Result: \( x = 7 \)

Example 2: Solving by Multiplication

Solve: \( \frac{x}{4} = 3 \)
1. We see "divided by 4". The opposite is "multiplied by 4".
2. Multiply both sides by 4: \( \frac{x}{4} \times 4 = 3 \times 4 \)
3. Result: \( x = 12 \)

Memory Trick: Think of the equal sign as a magic bridge. When a number crosses the bridge to the other side, it changes its sign to the opposite!

5. Handling Two-Step Equations

Sometimes, more than one thing is happening to the variable. For example: \( 2x + 3 = 11 \).

Step-by-Step Strategy:
1. Undo Addition or Subtraction first: Get rid of the \( +3 \) by subtracting 3 from both sides.
\( 2x + 3 - 3 = 11 - 3 \)
\( 2x = 8 \)
2. Undo Multiplication or Division second: Get rid of the 2 (which means \( 2 \times x \)) by dividing both sides by 2.
\( \frac{2x}{2} = \frac{8}{2} \)
\( x = 4 \)

Common Mistake to Avoid: Don't try to divide before you subtract! It’s usually much easier to move the "loose" numbers (the ones not attached to \( x \)) first.

6. From Words to Math (Word Problems)

In the Hong Kong Attainment Test, you will often need to turn a story into an equation. Look for these keywords:

Keywords for Addition \( + \): Sum, increased by, more than, total.
Keywords for Subtraction \( - \): Difference, decreased by, less than, remaining.
Keywords for Multiplication \( \times \): Product, times, twice (means \( \times 2 \)).
Keywords for Division \( \div \): Quotient, shared, per.
Keywords for the Equal Sign \( = \): Is, results in, becomes.

Example: "A number \( y \) increased by 5 is 12."
Translation: \( y + 5 = 12 \)

7. Final Tips for Success

1. Check your answer!
Once you find \( x \), put it back into the original equation. For \( x + 8 = 15 \), if you found \( x = 7 \), check: Does \( 7 + 8 = 15 \)? Yes! Now you know your answer is 100% correct.

2. Keep it neat.
Write your equations line by line. Keep the equal signs \( = \) lined up vertically so you don't get lost in your work.

3. Don't rush.
Read word problems twice. The first time to understand the story, and the second time to find the numbers and the variable.

Quick Summary Takeaway:
- A variable is a letter representing a mystery number.
- An equation is a balance scale with an \( = \) sign.
- Use opposite operations to move numbers across the "equal sign bridge."
- Always do the same thing to both sides!

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What is a variable in algebra?

A variable is a symbol (usually a letter like \( x \), \( y \), or \( a \)) that represents an unknown number.

What is the difference between an algebraic expression and an equation?

An algebraic expression is a group of terms without an equal sign, whereas an equation is a statement that two things are equal and always contains an \( = \) sign.

What is the Golden Rule of equations?

Whatever operation you perform on one side of an equation, you must perform the exact same operation on the other side.

What are opposite operations in algebra?

Opposite operations (or inverse operations) are pairs that undo each other: addition and subtraction are opposites, and multiplication and division are opposites.

How do you solve an equation involving addition?

To solve for a variable with a number added to it, subtract that same number from both sides of the equation.

How do you solve an equation involving division?

To solve for a variable that is being divided by a number, multiply both sides of the equation by that same number.

What is the general strategy for solving a two-step equation?

First, undo any addition or subtraction, then undo any multiplication or division to isolate the variable.

How do you translate word problems into equations?

Identify the variable and look for keywords like "sum" (addition), "difference" (subtraction), "product" (multiplication), and "is" (equal sign).

Why is it important to check your answer after solving an equation?

Checking your answer ensures that the value you found actually makes the original equation true and that you didn't make a calculation error.

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