A linear equation is an equation where the highest power of the variable is 1. Solving equations involves finding the value of the variable that makes the equation true. Coefficients may be whole numbers or fractions, and the same principles apply in each case.

 

 

Forming Linear Equations

Equations can be formed from expressions or from worded statements.

 

For example, the statement
“Three times a number plus five equals twenty”
can be written as:

$$
3x + 5 = 20
$$

Forming the equation correctly is the first step towards solving it.

 

 

Solving Linear Equations with Whole Number Coefficients

To solve a linear equation, use inverse operations to isolate the variable.

 

For example:

$$
3x + 5 = 20
$$

 

Subtract \( 5 \) from both sides:

$$
3x = 15
$$

 

Divide both sides by \( 3 \):

$$
x = 5
$$

 

Another example:

$$
7x - 4 = 17
$$

 

Add \( 4 \) to both sides:

$$
7x = 21
$$

 

Divide by \( 7 \):

$$
x = 3
$$

 

 

Solving Equations with the Variable on Both Sides

Sometimes the variable appears on both sides of the equation.

 

For example:

$$
5x + 2 = 2x + 14
$$

 

First collect the \( x \) terms on one side:

$$
5x - 2x = 14 - 2
$$

$$
3x = 12
$$

 

Divide by \( 3 \):

$$
x = 4
$$

 

 

Solving Equations with Fractional Coefficients

Equations may contain fractions as coefficients. These can be solved by clearing fractions or working carefully with inverse operations.

 

For example:

$$
\frac{x}{3} + 2 = 6
$$

 

Subtract \( 2 \) from both sides:

$$
\frac{x}{3} = 4
$$

 

Multiply both sides by \( 3 \):

$$
x = 12
$$

 

Another example:

$$
\frac{2x}{5} - 1 = 3
$$

 

Add \( 1 \) to both sides:

$$
\frac{2x}{5} = 4
$$

 

Multiply both sides by \( 5 \):

$$
2x = 20
$$

 

Divide by \( 2 \):

$$
x = 10
$$

 

 

Solving Simple Equations with Fractions on Both Sides

For example:

$$
\frac{x}{4} + \frac{1}{2} = \frac{3}{2}
$$

 

Subtract \( \frac{1}{2} \) from both sides:

$$
\frac{x}{4} = 1
$$

 

Multiply both sides by \( 4 \):

$$
x = 4
$$

 

 

Key Points to Remember

Carry out the same operation on both sides of the equation.
Use inverse operations in a logical order.
Be careful with fractions and signs.
Check the solution by substituting it back into the original equation.

 

Being confident with forming, manipulating and solving linear equations is essential for algebra and for many applications in mathematics and science.

---

Topic Revision Checklist

Equations and Inequalities – Algebraic Methods - Learning Objectives Checklist

---