To solve complicated linear equations, we use something called inverse operations.

This means we do the opposite of the operations we see in the equation.

Our goal is to get the variable (often called x) all by itself on one side of the equation.

The main operations we deal with are:

• Addition
• Subtraction
• Multiplication
• Division

How to Solve Linear Equations with Inverse Operations:

1. Look at the equation: Let's say we have $3x + 7 = 22$.

2. Use inverse operations:

   • First, subtract the number (which is called a constant) from both sides to get rid of it:

     $$3x + 7 - 7 = 22 - 7 \implies 3x = 15$$

   • Next, divide by the number in front of the variable (called the coefficient):

     $$\frac{3x}{3} = \frac{15}{3} \implies x = 5$$

3. Check your answer: Plug $x = 5$ back into the original equation to see if it works:

   $$3(5) + 7 = 22 \implies 15 + 7 = 22$$

   This shows our answer is correct!

Why Inverse Operations Are Important:

• Accuracy: Using inverse operations helps reduce mistakes. It makes sure that the answers we find are reliable.

• Efficiency: Inverse operations give us a clear way to solve tricky equations. This helps students work through problems step by step.

• Builds a Base for More Math: Learning how to use inverse operations is really important. It helps when we need to solve more complex things later, like quadratic equations, which are common in higher-level math.

In GCSE Mathematics, a high number of students (97%) said that understanding inverse operations helped them become better problem solvers. This shows just how valuable they are in learning math!