Practical Applications of Simultaneous Equations (VCE SSCE General Mathematics): Revision Notes

Worked Example: Finding Individual Flower Prices

Problem: Mark purchases 3 roses and 2 gardenias for $15.50. Peter purchases 5 roses and 3 gardenias for $24.50. What is the cost of each type of flower?

Solution:

Defining variables:

Let $r$ represent the cost of one rose (in dollars)

Let $g$ represent the cost of one gardenia (in dollars)

Setting up equations:

From Mark's purchase:

$3r + 2g = 15.5 \quad \text{(1)}$

From Peter's purchase:

$5r + 3g = 24.5 \quad \text{(2)}$

Using a CAS calculator:

To solve these equations using a CAS calculator, you'll need to enter both equations into the system. The calculator interface will look similar to this:

After entering the equations, access the solve function (often found in the Variables menu) to find the solution:

The calculator gives us:

$r = 2.5$

$g = 4$

Checking the solution:

Substitute $r = 2.5$ and $g = 4$ into equation (2):

$\text{LHS} = 5(2.5) + 3(4)$

$= 12.5 + 12$

$= 24.5 = \text{RHS} \checkmark$

The solution checks out correctly.

Final answer:

Roses cost $2.50 each and gardenias cost $4.00 each.

Worked Example: Finding Rectangle Dimensions

Problem: A rectangle has a perimeter of 48 cm. The length of the rectangle is three times the width. Find the dimensions of the rectangle.

Solution:

Defining variables:

Let $w$ represent the width of the rectangle (in cm)

Let $l$ represent the length of the rectangle (in cm)

Setting up equations:

The perimeter of a rectangle is the total distance around its edges. This can be calculated by adding together two widths and two lengths:

$2w + 2l = 48 \quad \text{(1)}$

We're told the length is three times the width:

$l = 3w \quad \text{(2)}$

Solving the system:

Use your CAS calculator to solve these simultaneous equations. Enter both equations into the solver function as demonstrated in Worked Example 1.

The solution is:

$w = 6$

$l = 18$

Final answer:

The rectangle has a width of 6 cm and a length of 18 cm.

Worked Example: Calculating Ticket Sales

Problem: Cinema tickets cost $19.50 for adults and $14.50 for children. A total of 200 tickets were sold, generating $3265 in revenue. How many children's tickets were sold?

Solution:

Defining variables:

Let $a$ represent the number of adult tickets sold

Let $c$ represent the number of children's tickets sold

Setting up equations:

The total revenue from ticket sales is:

$19.5a + 14.5c = 3265 \quad \text{(1)}$

The total number of tickets sold is:

$a + c = 200 \quad \text{(2)}$

Note: When you know a total quantity (like total tickets), this often provides your second equation.

Solving the system:

Use your CAS calculator to solve these simultaneous equations following the method shown in the previous examples.

The solution is:

$a = 73$

$c = 127$

Final answer:

127 children's tickets were sold.

Verification: You should also be able to work out that 73 adult tickets were sold (since $200 - 127 = 73$). You can verify by calculating: $19.5 \times 73 + 14.5 \times 127 = 1423.50 + 1841.50 = 3265$ ✓