Solving Linear Equations: Step-by-Step Guide for Singapore Students

The fundamental concept in equation solving is maintaining balance. Imagine an equation as a perfectly balanced scale. The equals sign ($=$) represents the fulcrum, the point on which the scale balances. To keep the scale balanced, any operation performed on one side must also be performed on the other.

For example, if you have the equation $x + 3 = 7$, you can subtract $3$ from both sides to isolate $x$. This gives you $x + 3 - 3 = 7 - 3$, which simplifies to $x = 4$. The scale remains balanced because you performed the same operation on both sides.

Expert Tip: Always double-check your work by substituting the solution back into the original equation. If both sides are equal, your solution is correct. This verification step is crucial for exam success and is emphasized in our Additional Math tuition classes.

Equations involving fractions can seem daunting, but there's a straightforward way to simplify them: eliminate the denominators. This is achieved by multiplying every term on both sides of the equation by the least common multiple (LCM) of the denominators.

Let's say you have the equation: $\frac{x}{2} + \frac{1}{3} = \frac{5}{6}$. The LCM of $2$, $3$, and $6$ is $6$. Multiply each term by $6$:

$(6 \times \frac{x}{2}) + (6 \times \frac{1}{3}) = (6 \times \frac {5}{6})$

This simplifies to: $3x + 2 = 5$. Now you have a much easier equation to solve.

Real-World Application: This technique is commonly used in recipes when scaling ingredients up or down. You're essentially solving an equation to determine the correct amount of each ingredient.

Once you've eliminated fractions (if present), the next step is to combine like terms on each side of the equation. Like terms are those that have the same variable raised to the same power (e.g., $3x$ and $-5x$) or are constants (e.g., $2$ and $-7$).

Consider the equation: $2x + 5 - x + 3 = 10$. Combine the '$x$' terms ($2x - x = x$) and the constant terms ($5 + 3 = 8$). This simplifies the equation to: $x + 8 = 10$.

Expert Tip: Pay close attention to the signs ($+$ or $-$) when combining terms. A common mistake is to incorrectly add or subtract negative numbers. If you're struggling with algebraic manipulation, our H2 Math tutors in Singapore can provide personalized guidance to strengthen your foundation.

The final step is to isolate the variable on one side of the equation. This means getting the variable term by itself, with a coefficient of $1$. To do this, use inverse operations. Addition and subtraction are inverse operations, as are multiplication and division.

Continuing with our example, $x + 8 = 10$, we need to isolate '$x$'. The inverse operation of adding $8$ is subtracting $8$. Subtract $8$ from both sides: $x + 8 - 8 = 10 - 8$. This gives us $x = 2$.

Real-World Application: This concept is used when budgeting. If you know your total income and expenses, you can isolate the variable (savings) to determine how much money you have left to save.

Here's a step-by-step guide to solving linear equations:

🎥 Watch this quick video tutorial on solving linear equations!

Step-by-Step Process:

1. Simplify both sides of the equation by eliminating parentheses and combining like terms
2. Eliminate fractions (if present) by multiplying through by the LCM
3. Move all variable terms to one side of the equation
4. Move all constant terms to the other side
5. Isolate the variable by dividing or multiplying both sides
6. Verify your solution by substituting back into the original equation

Need more practice? Try our free resources for additional worked examples and practice problems aligned with Singapore's O-Level and A-Level syllabuses.

Question: What if the variable appears on both sides of the equation?

Answer: Use addition or subtraction to move all terms containing the variable to one side of the equation. Then, proceed with the steps for isolating the variable.

Question: What if there is no solution to the equation?

Answer: If, after simplifying, you arrive at a statement that is always false (e.g., $2 = 5$), then the equation has no solution. This indicates the equation is inconsistent.

Question: How can I avoid making careless mistakes?

Answer: Work systematically through each step, write neatly, and always verify your answer. If you're consistently making errors, consider seeking help through Additional Math tuition or our trial lesson to identify and correct your misconceptions.