Solving Equations after Substitution Revision Notes for HSC SSCE Mathematics Standard

Solving Equations after Substitution

Understanding formulas

A formula is a mathematical relationship that connects two or more variables together. Formulas are used throughout mathematics and science to show how different quantities relate to each other.

Key term: A variable is a symbol (usually a letter) that represents a quantity that can change or vary.

Common examples of formulas

Here are two important formulas you should recognise:

Pythagoras' theorem:

$a^2 = b^2 + c^2$

This formula relates the three sides of a right-angled triangle. The variables are $a$ , $b$ , and $c$ , which represent the lengths of the sides.

Area of a circle:

$A = \pi r^2$

This formula connects the area and radius of a circle. The variables are $A$ (area) and $r$ (radius).

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These are two of the most commonly used formulas in mathematics. Being able to recognise and work with them fluently will help you in many different areas of your studies, from geometry to physics.

Using substitution with formulas

When you know the values of some variables in a formula, you can substitute these values into the formula to find an unknown variable. This process involves replacing the variable letters with the actual numbers you've been given.

However, there's an important step to consider: if the unknown variable is not the subject of the equation (meaning it's not by itself on one side of the equals sign), you'll need to rearrange and solve the equation first.

Example: In the formula $A = \pi r^2$ , the variable $A$ is the subject because it stands alone on the left side. If you needed to find $r$ instead, you would need to rearrange the equation to make $r$ the subject.

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Always check if the unknown variable is the subject first!

A common mistake is to substitute values before checking whether the unknown variable is isolated on one side of the equation. If it's not the subject, you must rearrange the formula before substituting values. This can save you from making calculation errors and getting confused mid-solution.

Step-by-step process for using a formula

Follow these five steps when working with formulas:

Step 1: Write the formula

Begin by clearly writing out the formula you need to use. This helps you identify which variables you're working with.

Step 2: Replace the variables with known values

Substitute the numbers given in the question for the corresponding variables in the formula. Be careful to match each number with the correct variable.

Step 3: Solve the equation if necessary

Check whether the unknown variable is the subject of the equation. If it isn't, you'll need to rearrange and solve the equation to isolate the unknown variable.

Step 4: Evaluate using a calculator

Use your calculator to work out the numerical value. Make sure you enter the calculation correctly, paying attention to the order of operations.

Step 5: Write your answer with correct units and accuracy

Express your final answer to the specified level of accuracy (such as a certain number of decimal places) and always include the appropriate units of measurement.

Worked example: Finding the radius of a traffic cone

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Worked Example: Calculating the Radius of a Traffic Cone

Problem: A traffic cone has a volume of 4000 cubic centimetres and a height of 40 centimetres.

Use the formula $r = \sqrt{\frac{3V}{\pi h}}$ to find the radius of the base of the cone. Give your answer correct to two decimal places.

Solution:

Step 1: Write the formula.

$r = \sqrt{\frac{3V}{\pi h}}$

Step 2: Substitute the known values into the formula.

We know that $V = 4000$ and $h = 40$ . Replacing these values gives us:

$r = \sqrt{\frac{3 \times 4000}{\pi \times 40}}$

Step 3: In this case, $r$ is already the subject of the formula, so we can proceed straight to evaluation.

Step 4: Evaluate the expression.

$r = \sqrt{\frac{3 \times 4000}{\pi \times 40}}$

$r = 9.772050238...$

Step 5: Round to two decimal places and include units.

$r \approx 9.77 \text{ cm}$

The radius of the base of the cone is approximately 9.77 cm.

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Exam tip: Always check your formula carefully before substituting values. A common mistake is to substitute numbers into the wrong variables. Also, remember to show your working - even if you use a calculator, write down the steps you're following.

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Key Points to Remember:

A formula is a mathematical relationship between two or more variables
Substitution means replacing variables with their known numerical values
Always check if the unknown variable is the subject of the equation before starting your calculation
Follow the five-step process: Write, Replace, Solve (if needed), Evaluate, and Write your answer
Include correct units and round to the specified accuracy in your final answer

Solving Equations after Substitution (HSC SSCE Mathematics Standard): Revision Notes