Alliance Form 3 P1 Q3 — Simultaneous Equations (Technicians & Artisans)

KCSE (Form 3) Form 3 Algebra

Published

The Question

“A company hires technicians and artisans. On one job, three technicians and two artisans cost 9,000 shillings. On another job, four technicians and one artisan cost 9,500 shillings. Find the total cost of hiring two technicians and five artisans.”

1

Introduce letters for the unknowns

Word problems become manageable once you name the quantities you do not know. Let X stand for the cost of hiring one technician and Y for the cost of hiring one artisan. Every sentence in the question can now be written as an equation in X and Y.

x=cost of one technician,y=cost of one artisanx = \text{cost of one technician}, \quad y = \text{cost of one artisan}
2

Form the two equations

The first job describes three technicians and two artisans costing 9,000, and the second job describes four technicians and one artisan costing 9,500. Translating each job word for word gives two equations in the two unknowns, which is exactly what we need to solve for both.

3x+2y=9000(equation 1)3x + 2y = 9000 \quad \text{(equation 1)}
4x+y=9500(equation 2)4x + y = 9500 \quad \text{(equation 2)}
3

Eliminate one unknown

To use the elimination method we make the coefficients of one letter match. Equation two has a single Y, so multiplying the whole of equation two by two gives a 2y term, matching equation one. Subtracting equation one from this new equation cancels the Y terms and leaves a single equation in X.

2×(4x+y=9500)⇒8x+2y=190002 \times (4x + y = 9500) \Rightarrow 8x + 2y = 19000
(8x+2y)−(3x+2y)=19000−9000(8x + 2y) - (3x + 2y) = 19000 - 9000
5x=100005x = 10000
4

Solve for the first unknown

With only X remaining, divide both sides by five. This gives the cost of hiring one technician.

x=100005=2000x = \frac{10000}{5} = 2000

So a technician costs 2,000 shillings.

5

Substitute back to find the second unknown

Now that X is known, put it back into either original equation. Equation two is simplest because it has only one Y. Replacing X with 2,000 and rearranging gives the cost of one artisan.

4(2000)+y=95004(2000) + y = 9500
8000+y=95008000 + y = 9500
y=9500−8000=1500y = 9500 - 8000 = 1500

So an artisan costs 1,500 shillings.

6

Answer the actual question

The question does not ask for X and Y on their own; it asks for the cost of two technicians and five artisans. Substitute the values you found into that expression and add the two amounts.

2x+5y=2(2000)+5(1500)2x + 5y = 2(2000) + 5(1500)
=4000+7500=11500= 4000 + 7500 = 11500

Final Result

Hiring two technicians and five artisans costs 11,500 shillings (a technician is 2,000 shillings and an artisan is 1,500 shillings).

Why this method works

The method works because each real-life job gives a genuine relationship between the two costs, and two independent relationships are enough to pin down two unknowns exactly. Elimination exploits this by scaling one equation so a shared term matches, then subtracting to remove that term — turning a two-variable problem into a one-variable one that is easy to solve. Once one cost is known, substitution recovers the other, and the final expression is just a matter of plugging the confirmed prices into the combination the question asked about.

✓

Check with equation 1: 3(2000) + 2(1500) = 6000 + 3000 = 9000, which matches the first job.