Linear Equations Revision Notes for HSC SSCE Mathematics Standard

Linear Equations

Understanding equations

An equation is a mathematical statement showing that two expressions are equal. It always contains an equals sign. For example, $x + 4 = 7$ is an equation.

In a linear equation, all variables are raised to the power of $1$ . This means you won't see terms like $x^2$ or $x^3$ in linear equations. The process of finding the value of the unknown variable is called solving the equation.

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Think of linear equations as the simplest type of equation - they create straight lines when graphed. The "linear" in their name comes from the fact that variables are never squared, cubed, or raised to any power other than 1.

Key concepts for solving equations

When solving equations, we use opposite operations to isolate the variable:

Addition $(+)$ is the opposite of subtraction $(-)$
Multiplication $(\times)$ is the opposite of division $(\div)$

The most important principle is that equations must stay balanced. Whatever operation you perform on one side of the equals sign, you must perform on the other side as well. Think of it like a set of scales - both sides need to stay equal.

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The Golden Rule of Equations: Whatever you do to one side of the equation, you MUST do to the other side. This keeps the equation balanced and maintains equality. Breaking this rule will lead to incorrect answers!

The goal when solving an equation is to get the pronumeral (variable) by itself on one side of the equation.

Steps for solving equations

To solve any linear equation, follow these steps:

Step 1: Identify the opposite operation needed (remember: $+$ is opposite to $-$ , and $\times$ is opposite to $\div$ )

Step 2: Add or subtract the same number from both sides of the equation

Step 3: Multiply or divide both sides of the equation by the same number

Step 4: For two-step, three-step or four-step equations, repeat steps 1 to 3 as required. It's usually easier to deal with addition and subtraction first, then handle multiplication and division.

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Strategy Tip: When faced with complex equations, always tackle addition and subtraction operations before moving on to multiplication and division. This makes the process smoother and reduces the chance of errors.

Checking your solution

Once you've found a solution, always verify it's correct. Substitute your answer back into the original equation. If the left-hand side equals the right-hand side, your solution is correct.

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Always Check Your Answer! Substituting your solution back into the original equation is not optional - it's a crucial step that confirms you haven't made any errors along the way. If both sides don't equal after substitution, you need to rework the problem.

Worked example: Two-step linear equation

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Worked Example: Solving a Two-Step Equation

Let's solve the equation $3x + 5 = -2$ .

Step 1: Write the equation

$3x + 5 = -2$

Step 2: The opposite of adding $5$ is subtracting $5$ . Subtract $5$ from both sides

$3x + 5 - 5 = -2 - 5$

$3x = -7$

Step 3: The opposite of multiplying by $3$ is dividing by $3$ . Divide both sides by $3$

$\frac{3x}{3} = \frac{-7}{3}$

$x = -\frac{7}{3}$

Step 4: Express as a mixed number

$x = -2\frac{1}{3}$

Step 5: Check the solution by substituting $x = -2\frac{1}{3}$ back into the original equation to verify both sides are equal.

Worked example: Linear equation with fractions

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Worked Example: Solving an Equation with Fractions

Let's solve the equation $\frac{y}{3} - \frac{y}{7} = 2$ . Express the answer as a simple fraction.

Step 1: Write the equation

$\frac{y}{3} - \frac{y}{7} = 2$

Step 2: To simplify the fractions, find the lowest common denominator. The smallest number divisible by both $3$ and $7$ is 21

Step 3: Multiply both sides of the equation (all terms) by $21$

$21 \times \left(\frac{y}{3}\right) - 21 \times \left(\frac{y}{7}\right) = 2 \times 21$

Step 4: Cancel out the common factors ( $21$ divided by $3$ is $7$ , and $21$ divided by $7$ is $3$ )

Step 5: Write the equation without fractions

$7y - 3y = 42$

Step 6: Subtract the like terms ( $7y - 3y = 4y$ )

$4y = 42$

Step 7: The opposite of multiplying by $4$ is dividing by $4$ . Divide both sides by $4$

$\frac{4y}{4} = \frac{42}{4}$

Step 8: Express as a mixed number in simplest form

$y = 10\frac{1}{2}$

Step 9: Check the solution by substituting $y = 10\frac{1}{2}$ into the original equation to verify it's correct.

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When dealing with fractions in equations, your first goal should be to eliminate them by multiplying through by the lowest common denominator. This transforms the equation into a simpler form without fractions, making it much easier to solve.

Worked example: Four-step linear equation

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Worked Example: Solving a Four-Step Equation

Let's solve the equation $\frac{b}{2} + 5 = 3b + 7$ .

Step 1: Write the equation

$\frac{b}{2} + 5 = 3b + 7$

Step 2: To simplify the fraction, find the lowest common denominator. The only denominator is $2$

Step 3: Multiply both sides of the equation (all terms) by $2$

$\left(\frac{b}{2}\right) \times 2 + 5 \times 2 = 3b \times 2 + 7 \times 2$

Step 4: Cancel out the common factor ( $2$ divided by $2$ is $1$ )

Step 5: Write the equation without a fraction

$b + 10 = 6b + 14$

Step 6: The opposite of adding $6b$ is subtracting $6b$ . Subtract $6b$ from both sides

$b + 10 - 6b = 6b + 14 - 6b$

$-5b + 10 = 14$

Step 7: The opposite of adding $10$ is subtracting $10$ . Subtract $10$ from both sides

$-5b + 10 - 10 = 14 - 10$

$-5b = 4$

Step 8: The opposite of multiplying by $-5$ is dividing by $-5$ . Divide both sides by $-5$

$\frac{-5b}{-5} = \frac{4}{-5}$

Step 9: Express as a fraction in simplest form

$b = -\frac{4}{5}$

Step 10: Check that the solution is correct by substituting back into the original equation.

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Working with Variables on Both Sides: When you have variables on both sides of the equation (like in this example), your first step after eliminating fractions should be to collect all variable terms on one side. Choose the side that will give you a positive coefficient to make the rest of the work easier.

Exam tips

Approaching linear equations in exams requires both mathematical skill and good presentation. Here are some key strategies for success:

Always show your working clearly, writing each step on a new line
When dealing with fractions, identify the lowest common denominator before multiplying through
Remember to perform operations on all terms on both sides of the equation
Check your final answer by substituting it back into the original equation
Express your final answer in the form requested (simple fraction, mixed number, decimal, etc.)

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Common Mistakes to Avoid:

Forgetting to multiply ALL terms when eliminating fractions
Not performing the same operation on both sides of the equation
Making sign errors when dealing with negative numbers
Failing to simplify your final answer to the requested form

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Key Points to Remember:

A linear equation has all variables raised to the power of $1$
Use opposite operations to solve equations: addition undoes subtraction, and multiplication undoes division
Always perform the same operation on both sides to keep the equation balanced
It's usually easier to deal with addition/subtraction before multiplication/division
Check your solution by substituting your answer back into the original equation
When working with fractions, multiply through by the lowest common denominator to eliminate them

Linear Equations (HSC SSCE Mathematics Standard): Revision Notes