IP EMaths Notes (Upper Sec, Year 3-4): 05) Simultaneous Equations and Word Problems

Simultaneous equations convert real-world statements into solvable algebra. Choose the method - substitution, elimination, or graphical - that minimises manipulation.

Strategy snapshot

• Rearrange one equation to make a variable the subject for substitution.
• In elimination, line up coefficients so one variable cancels cleanly.
• Always interpret the algebraic solution back in context to check feasibility.

Worked example - Ticket mix at a concert

Adult tickets cost \(\$48\) and student tickets cost \(\$32\). At a weekend concert, 360 tickets are sold for a total of \(\$14400\). How many of each ticket type were sold?

1. Let \(a\) be the number of adult tickets and \(s\) the number of student tickets. Translate the statements:
   • \(a + s = 360\).
   • \(48a + 32s = 14400\).
2. Use elimination. Multiply the first equation by \(32\): \(32a + 32s = 11520\).
3. Subtract this from the revenue equation: \((48a + 32s) - (32a + 32s) = 14400 - 11520\) giving \(16a = 2880\).
4. Solve for \(a\): \(a = \dfrac{2880}{16} = 180\).
5. Substitute back into \(a + s = 360\): \(180 + s = 360 \Rightarrow s = 180\).
6. Quick check: \(48 \times 180 + 32 \times 180 = 8640 + 5760 = 14400\) which matches the stated revenue.

Therefore 180 adult tickets and 180 student tickets were sold.

Try this

Solve \(\begin{cases}2x + 3y = 19\\x - y = 4\end{cases}\) and explain what \(x\) and \(y\) represent if the equations model counts of two bundle types in a fund-raising sale.