Solving Linear Equations

Table of Contents
1. What are linear equations?
2. How do I solve a linear equation?
3. How do I solve a linear equation of the form ax + b = c?
4. How to Solve a Linear Equation with X on Both Sides
5. How do I solve a linear equation that contains brackets?
6. How do I solve a linear equation that contains fractions?
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What are linear equations?

• A linear equation is an equation that results in a straight line when graphed.
• The highest power in a linear equation is 1, indicating no terms of x² or higher.
• A standard linear equation takes the form ax + b = c, where a, b, and c are constants, and x is the variable.

How do I solve a linear equation?

• To solve a linear equation, isolate the variable, usually $𝑥$x, by performing inverse operations on both sides of the equation.
  • Inverse operations are the opposite of the operations previously applied to the variable.
• The order in which inverse operations are performed is important.
  • Generally, this will follow BIDMAS (Brackets, Indices, Division/Multiplication, Addition/Subtraction) in reverse order.
  • However, the exact sequence depends on the order in which the original operations were applied to the variable to form the equation.

How do I solve a linear equation of the form ax + b = c?

• The operations that have been applied to x here are:
  • STEP 1: Multiply by a
  • STEP 2: Add b
• To solve this, you must carry out the inverse operations in reverse order:
  • STEP 1: Subtract b
  • STEP 2: Divide by a
• For example, to solve the equation 2x + 1 = 9:
  • STEP 1: Subtract 1
    
  • STEP 2: Divide by 2
    
• Be extra careful if any of the terms have negatives. For example, to solve the equation 2 - 3x = 10: 
  • STEP 1: Subtract 2
    
  • STEP 2: Divide by -3 (be careful not to drop the negative sign)
    

How to Solve a Linear Equation with X on Both Sides

• When dealing with a linear equation that has the unknown variable 'x' on both sides, the first step is to gather all terms with 'x' on one side of the equation. This simplifies the process.
• It's usually easier to move the term with the smallest coefficient to the other side. This means shifting the term with the lowest number in front of 'x'.
• For example, to solve the equation 4x - 7 = 11 + x: 
  • STEP 1: Move the 𝑥 x term with the smallest coefficient of x. The coefficients are 4 and 1, so move the x-term on the right-hand side:
    
  • STEP 2: Solve the linear equation using the method above:
    3x - 7 = 11
    3x = 18
    x = 6

How do I solve a linear equation that contains brackets?

• If a linear equation contains brackets on one or both sides, start by expanding the brackets. 
• For example, to solve the equation 2(x - 1) = 10 - (3 + x):
  • STEP 1: Expand the brackets on both sides:
    2x - 2 = 10 - 3 - x
  • STEP 2: Collect the x-terms to one side by subtracting the term with the smaller coefficient of x:
    Be extra careful if any of the terms have negatives
  • STEP 3: Solve the linear equation using the method above:
    3x - 2 = 7
    3x = 9
    x = 3

How do I solve a linear equation that contains fractions?

• If a linear equation contains a fraction on one or both sides, remove the fractions by multiplying both sides by everything on the denominator. 
  • Remember to put brackets around the expression first.
• For example, to solve the equation you will need to multiply both sides of the equation by both (x + 3) and (4 - 2x)
• STEP 1: Multiply both sides by (x + 3)
  
• STEP 2: Multiply both sides by (4 - 2x)
  
• STEP 3: Expand the brackets on both sides
   
• STEP 4: Collect the x terms to one side by subtracting the term with the smaller coefficient of x
  
• STEP 5: Solve the equation: You can swap the sides if it makes solving the equation easier