Solving Linear Equations and Developing Formulas (VCE SSCE General Mathematics): Revision Notes

Worked Example: Simple Addition Equation

Solve $x + 6 = 10$

Step 1: Write the equation

$x + 6 = 10$

Step 2: Use the inverse operation to isolate $x$

Since $6$ is being added to $x$, we subtract $6$ from both sides:

$x + 6 - 6 = 10 - 6$

$\therefore x = 4$

Step 3: Check the solution

Substitute $x = 4$ back into the original equation:

$\text{LHS} = x + 6$

$= 4 + 6$

$= 10$

$= \text{RHS}$

$\therefore$ Solution is correct.

Worked Example: Equation with Multiplication

Solve $3y = 18$

Step 1: Write the equation

$3y = 18$

Step 2: Use the inverse operation

Since $y$ is being multiplied by $3$, we divide both sides by $3$:

$\frac{3y}{3} = \frac{18}{3}$

$\therefore y = 6$

Step 3: Check the solution

$\text{LHS} = 3y$

$= 3 \times 6$

$= 18$

$= \text{RHS}$

$\therefore$ Solution is correct.

Worked Example: Equation with Brackets

Solve $4(x - 3) = 24$

There are two methods for solving equations with brackets:

Method 1: Expand first

Step 1: Write the equation

$4(x - 3) = 24$

Step 2: Expand the brackets

$4x - 12 = 24$

Step 3: Add $12$ to both sides

$4x - 12 + 12 = 24 + 12$

$4x = 36$

Step 4: Divide both sides by $4$

$\frac{4x}{4} = \frac{36}{4}$

$\therefore x = 9$

Method 2: Divide first

Step 1: Write the equation

$4(x - 3) = 24$

Step 2: Divide both sides by $4$

$\frac{4(x - 3)}{4} = \frac{24}{4}$

$x - 3 = 6$

Step 3: Add $3$ to both sides

$x - 3 + 3 = 6 + 3$

$\therefore x = 9$

Check the solution:

$\text{LHS} = 4(x - 3)$

$= 4(9 - 3)$

$= 4 \times 6 = 24 = \text{RHS}$

$\therefore$ Solution is correct.

Both methods arrive at the same answer. Choose the method you find most comfortable.

Worked Example: Using the solve() Function

To solve $-4 - 5b = 8$:

Step 1: Enter the solve command on your CAS calculator

Type: solve(-4 - 5b = 8, b)

This tells the calculator to solve the equation for variable $b$.

Step 2: Read the result

$b = -2.4$

Worked Example: Geometric Problems

Problem: Find an equation for the perimeter of a triangle with sides $4$, $7$, and $x$.

Solution:

Step 1: Choose a variable for perimeter

Let $P$ be the perimeter.

Step 2: Add all sides of the triangle

Remember that perimeter is the total distance around the outside of a shape.

$P = 4 + 7 + x$

Step 3: Simplify

$P = 11 + x$

Answer: The required equation is $P = 11 + x$

Worked Example: Number Problems

Problem: If $11$ is added to a certain number, the result is $25$. Find the number.

Solution:

Step 1: Choose a variable for the unknown

Let $n$ be the number.

Step 2: Write an equation from the information

"$11$ is added to a number" gives us: $n + 11$

"The result is $25$" means: $n + 11 = 25$

Step 3: Solve the equation

Subtract $11$ from both sides:

$n + 11 - 11 = 25 - 11$

$\therefore n = 14$

Answer: The required number is $14$.

Worked Example: Practical Application

Problem: A car rental company charges a fixed fee of $110 plus $84 per day. The Brown family have budgeted $650 for car hire during their holiday. For how many days can they hire a car?

Solution:

Step 1: Define the variable and set up the equation

Let $d$ be the number of days the car is hired.

Total cost = Fixed charge + (Daily rate × Number of days)

$110 + 84d = 650$

Step 2: Solve the equation

Subtract $110$ from both sides:

$110 + 84d - 110 = 650 - 110$

$84d = 540$

Divide both sides by $84$:

$\frac{84d}{84} = \frac{540}{84}$

$\therefore d = 6.428...$

Step 3: Interpret the answer in context

Car hire operates on complete days only. Since the family cannot hire for part of a day, and they must stay within their $650 budget, we round down to $6$ days.

If they hired for $7$ days, the cost would be: 110 + 84(7) = $698$, which exceeds their budget.

Answer: The Brown family can hire a car for $6$ days.