Download Solving Linear Equations - A Mathematical Mischief Tutorial

Mathematical Mischief (2012)
Solving Linear Equations
All over the world, people are using maths to solve simple problems. Things like
calculating the cost of electrical work, food quantities, etc.
In some cases, we use linear equations to define the functions we use for these. They’re
generally defined in a fairly basic way, that is:
y = mx + c
Now, let’s say we know what y is equal to, how do we find what x is equal to?
Now, there’s a simple process we need to apply to this. It’s called inverse operations, or
reverse BODMAS (SAMDOB). Basically, it’s done in 3 steps:
•Expand brackets, and collect like terms.
•Identify how the equation is made up.
•Reverse the operations that are made, to isolate x . This must be done to both
sides of the equation, in order to keep the equation balanced.
Then, we state the value of x .
Let’s apply this with our first example, solving y = mx + c for x .
y = mx + c
y= m×x+c
y − c = mx
y−c
=x
m
Step 1. Write down the equation.
Step 2. Identify how the equation is made up.
Step 3. Reverse the operations that are made to isolate x .
First, we minus c from both sides of the equation.
Step 4. Divide both sides of the equation by m .
Now, this may seem way too advanced to do immediately. If you can’t understand it, try
something simpler.
Josh Young - Mathematical Mischief
Mathematical Mischief (2012)
Example 2: Identify the order of operations on the equation y = 3x + 9 .
y = 3x + 9
Step 1: Write down the equation.
Step 2: Take x .
x
Step 3: Multiply x by 3 (Multiplication).
3x
3x + 9
Step 4: Add 9.
Step 4: State the order of operations.
The order of operations for the equation for the equation y = 3x + 9 is the following:
•Take x .
•Multiply by 3.
•Add 9 to the equation.
Once you’ve gained sufficient practice, you don’t need to identify the order of operations.
For starters, though, it would be wise to practice identifying how the equations are built up,
in order to have a better understanding of taking them down, when solving for x .
Now, let’s apply this to another equation, and solve for x .
Example 3: Identify the order of operations on the equation 1 = 3(x + 2) . Hence solve for x .
1 = 3(x + 2)
Step 1: Write down the equation.
Step 2: Take x .
x
x+2
3(x + 2)
Step 3: Add 2 to x (Brackets).
Step 4: Multiply by 3 (Multiplication).
Step 5: State the order of operations.
The order of operations for the equation for the equation 3(x + 2) is the following:
•Take x .
•Add 2.
•Multiply by 3.
1 = 3(x + 2)
Step 6: Apply inverse operations on the equation.
Write down the equation, and divide both sides by 3.
1
= x+2
3
−5
=x
3
−5
=x
3
Josh Young - Mathematical Mischief
Step 7: Subtract 2 from both sides.
Step 8: State the value of x .
Mathematical Mischief (2012)
In the case where we have like terms, identifying the steps can be difficult. As an
alternative, we need to collect the terms, before identifying the steps to the solution.
Example 4: Take the equation x + 9 = 4x − 3 and solve for x .
Step 1: Write down the equation.
x + 9 = 4x − 3
Step 2: Subtract x .
9 = 3x − 3
Step 3: Add 3.
12 = 3x
Step 4: Divide by 3.
4=x
Example 5: Take the equation 3(x − 4) = 4(x + 3) . Solve for x .
Step 1: Write down the equation.
3(x − 4) = 4(x + 3)
Step 2: Expand both brackets.
3x − 12 = 4x + 12
Step 3: Subtract 3x .
−12 = x + 12
Step 4: Subtract 12.
−24 = x
What now?
Now, you should attempt Self Help Tutorial: Solving Linear Equations.
Josh Young - Mathematical Mischief