Solving Linear Equations

1. The Core Principle: Keeping the Balance

Think of an equation as a perfectly balanced set of scales. The equals sign (=) tells you that the left side has the exact same value as the right side. The fundamental rule of solving equations is the balance principle: whatever you do to one side of the equation, you must do to the other to maintain the balance.

Our goal is to isolate the variable (e.g., get $x$ on its own). We do this by "undoing" the operations that have been applied to it. This means using inverse operations:

• The inverse of addition is subtraction.
• The inverse of subtraction is addition.
• The inverse of multiplication is division.
• The inverse of division is multiplication.

Let's start with a basic equation that has the variable on both sides:
$5x+3=2x+15$

Our strategy is to get all the $x$ terms on one side and all the constant terms (the numbers) on the other. A good practice is to move the variable term with the smaller coefficient to avoid working with negatives.

1. Collect variable terms: The $2x$ term is smaller than $5x$, so let's subtract $2x$ from both sides to keep the balance. $$\begin{array}{rl}5x-2x+3& =2x-2x+15\\ 3x+3& =15\end{array}$$
2. Collect constant terms: Now, we need to move the +3. We do the inverse by subtracting 3 from both sides. $$\begin{array}{rl}3x+3-3& =15-3\\ 3x& =12\end{array}$$
3. Isolate the variable: The variable $x$ is being multiplied by 3. The inverse is division, so we divide both sides by 3. $$\begin{array}{rl}\frac{3x}{3}& =\frac{12}{3}\\ x& =4\end{array}$$

We've found our solution! If you substitute $x=4$ back into the original equation, you'll find that both sides equal 23.

2. Solving Equations with Brackets (Parentheses)

What if an equation involves brackets, like $4(x+1)=3x+10$?

This is where your skills from the last lesson come directly into play. The first step is always to simplify the equation, which means expanding the brackets.

Let's watch a short video that walks through this process. It uses the British terminology you're familiar with.

As the video shows, the general procedure is:

1. Expand all brackets using the distributive property.
2. Combine any like terms on each side of the equation (if necessary).
3. Solve the resulting equation by collecting variable terms on one side and constants on the other.

3. Solving Equations with Fractions

Fractions in equations can seem complex, but there is a very effective strategy to deal with them: eliminate the fractions right at the start.

To do this, you find the Lowest Common Multiple (LCM) of all the denominators in the equation. This is also known as the Least Common Denominator (LCD). Then, you multiply every single term in the equation by this LCM.

Let's watch a detailed video that breaks this method down into two simple steps: 1) Eliminate the fractions, and 2) Solve the simplified equation.

Key Ideas from the Video

• Main Strategy: Multiply every term by the LCM of the denominators. This cancels out the denominators and leaves you with a much simpler equation.
• Don't Forget Whole Numbers: You must also multiply any terms that are not fractions by the LCM to keep the equation balanced.
• Cross-Multiplication: This is a shortcut for the special case where you have one fraction equal to another fraction (a proportion). If $\frac{a}{b}=\frac{c}{d}$, then $ad=bc$.
• Prime Factorisation for LCM: For more difficult denominators, breaking them down into their prime factors is a reliable way to find the LCM.

For additional reference and more worked examples, you can also read through the following guide. It uses British terminology and provides excellent step-by-step solutions.

Conclusion

Fantastic work today! You have now learned the complete process for solving the most common type of algebraic equation. By combining the simplification skills from our last lesson with the balance principle, you can now systematically find the solution to a wide variety of problems.

Let's summarise the universal strategy:

1. Clear Fractions: If there are fractions, multiply every term by the LCM of the denominators first.
2. Expand Brackets: Use the distributive property to remove any brackets.
3. Collect Like Terms: Simplify each side of the equation by combining like terms.
4. Isolate the Variable: Use inverse operations to gather all variable terms on one side and all constant terms on the other.
5. Solve: Perform the final division or multiplication to find the value of the variable.
6. Check your answer: Substitute your solution back into the original equation to ensure both sides are equal. This is the best way to catch any errors.

In our next lesson, we will move on to solving linear inequalities. The good news is that you will use the exact same methods we learned today. There is just one new, crucial rule to learn when dealing with inequality signs like < and >.