Understanding Solving Linear Equations

What Are Linear Equations?

A linear equation is an equation where the highest power of the variable (e.g., $x$) is 1. It forms a straight line when plotted on a graph, but for now, we’re focusing on solving them step by step.

Think of solving an equation as balancing a set of scales: whatever you do to one side, you must do to the other side to keep the balance

 

How to Solve Linear Equations

Key Steps:

1. Simplify both sides: Expand any brackets and combine like terms.
2. Collect the variable to one side: Make sure all occurrences of the variable are collected on one side 
3. Isolate the variable: Use addition, subtraction, multiplication, or division (BIDMAS) to get the variable by itself.
4. Check your solution: Substitute your solution back into the original equation to see if it works.

 

Tip

When rearranging you can 'move' variables from one side to the other by applying the opposite operation to both sides, i.e $$2x - 11 = 13 \\ \text{To 'move' the 11 from the left hand side of the equation} // \text{ I can do the opposite by adding 11 to both sides} \\ \text{this cancels out the -11 on the left and adds it to the right to give} \\ 2x - 11 + 11 =  13 + 11 \\ 2x = 24$$

 

Examples

Example 1: Solve $3x + 5 = 20$

Step 1: Subtract 5 from both sides to remove the constant: $$3x + 5 - 5 = 20 - 5 \\  3x = 15$$

Step 2: Divide both sides by 3 to isolate $x$: $$x = \frac{15}{3}  \\  x = 5$$

Check: Substitute $x = 5$ into the original equation: $$3(5) + 5 = 20$$

 

 

Example 2: Solve $2x + 3  = 7x - 12$

Step 1: Subtract $2x$ from both sides to move all $x$-terms to one side: $$2x - 2x + 3 = 7x - 2x - 12  \\ 3 = 5x - 12$$

Step 2: Add 12 to both sides to remove the constant from the right: $$3 + 12 = 5x - 12 + 12 \\ 15 = 5x$$

Step 3: Divide both sides by 5: $$x = \frac{15}{5}  \\ x = 3$$

Check: Substitute $x = 3$ into the original equation: $$2(3) + 3 = 7(3) - 12 \\ 6 + 3 = 21 -12  \\ 9 = 9$$

 

 

 

 

Examples: Solving Complex Linear Equations

Example 3: Solve $5(x - 3) + 2 =  3(2x + 1) -4$

Step 1: Expand the brackets $$5x - 15 + 2 = 6x + 3 -4 \\ \text{Simplify both sides:}  \\ 5x - 13 = 6x -1$$

Step 2: Move all $x$-terms to one side. Subtract $5x$ from both sides: $$-13 = x -1$$

Step 3: Move constants to the other side. Add 1 to both sides: $$-12 = x$$

Solution: $x = -12$

 

 

 

 

 

Example 3: Solve $\frac{2x+5}{3} = \frac{x -1}{2} + 1$

Step 1: Eliminate fractions by multiplying through by the lowest common denominator (LCD), which is 6: $$6 \times \frac{2x + 5}{3} = 6 \times \frac{x -1}{2} + 6 \times 1 \\ \text{Simplify each term:}  \\ 2(2x + 5) = 3(x -1) + 6$$

Step 2: Expand the brackets. $$4x + 10 =  3x -3 + 6 \\ \text{Simplify both sides: } \\ 4x + 10 = 3x + 3$$

Step 3: Move $x$-terms to one side and constants to the other. Subtract $3x$ from both sides: $$x + 10 = 3 \\ \text{Subtract 10 from both sides:} \\ x = -7$$

Solution: $x = -7$