Solving Linear Equations
What are linear equations?
- A linear equation is an equation that will produce a straight line when plotted on a graph
- The greatest power of in a linear equation is 1
- This means there are no terms of or a higher order
- A linear equation is normally in form
- where and are constants and is a variable
- where and are constants and is a variable
How do I solve a linear equation?
- To solve a linear equation you need to isolate the variable, usually , by carrying out inverse operations to both sides of the equation
- Inverse operations are just the opposite operations to what has already happened to the variable
- The order in which the inverse operations are carried out is important
- Most of the time, this will be BIDMAS in reverse
- However it depends on the order in which the 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 here are:
- Multiply by then add
- To solve this, you must carry out the inverse operations in reverse order
- Subtract then divide by
- The order in which you perform the inverse operations is important
- For example, to solve the equation
- Subtract 1
-
- Divide by 2
- Be extra careful if any of the terms have negatives
- For example, to solve the equation
- Subtract 2
-
-
- Be careful not to drop the negative sign on -3
- Divide by -3
-
How do I solve a linear equation with the unknown variable, x, on both sides?
- If a linear equation contains the unknown variable, usually , on both sides start by collecting these terms together on one side of the equation
- Moving the term with the smallest coefficient (number in front of ) is easiest
- For example, to solve the equation
- STEP 1
Move the term with the smallest coefficient of- The coefficients are 4 and 1 so move the -term on the right hand side
- STEP 1
-
- STEP 2
Solve the linear equation using the method above
- STEP 2
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
- STEP 1
Expand the brackets on both sides
- STEP 1
-
- STEP 2
Collect the terms to one side by subtracting the term with the smaller coefficient of
- STEP 2
-
-
- Be extra careful if any of the terms have negatives
- STEP 3
Solve the linear equation using the method above
-
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 and
- STEP 1
Multiply both sides by
- STEP 1
-
- STEP 2
Multiply both sides by
- STEP 2
-
- STEP 3
Expand the brackets on both sides
- STEP 3
-
- STEP 4
Collect the terms to one side by subtracting the term with the smaller coefficient of
- STEP 4
-
- STEP 5
Solve the equation- You can swap the sides if it makes solving the equation easier
- STEP 5
Examiner Tip
- If you have time in the exam, you should substitute your answer back into the equation to check you got it right
Worked example
(a)
Solve the equation.
Get rid of the fraction by multiplying both sides by the denominator .
Expand the brackets.
Bring the x terms to one side of the equation by subtracting x from both sides.
Get the x term by itself by adding 2 to both sides.
Solve by dividing both sides by 2.
(b)
Solve the equation.
Isolate the fraction by subtracting 2 from both sides.
Get rid of the fraction by multiplying both sides by the denominator (3).
Get the x term by itself by subtracting 1 from both sides.