Simultaneous Linear Equations
Senior 2 · Mathematics
Simultaneous Linear Equations
Lesson 1 of 4
20 min
At Kimironko market, Uwase pays 4,900 Frw for 2 kg of beans and 3 kg of rice. Her neighbour pays 3,500 Frw for 3 kg of beans and 1 kg of rice. What is the price of one kilogram of each? One equation with two unknown prices cannot answer this, because many pairs of prices fit. Two equations together can.
Pairs of equations like this are called simultaneous equations. They are used to compare phone tariffs, mix products, plan budgets and, in science, to find two unknown quantities from two measurements.
A linear equation in two variables has the form ax + by = c, where a, b and c are numbers and a and b are not both zero. Two such equations considered together form a system of simultaneous linear equations:
a1x + b1y = c1
a2x + b2y = c2
A solution of the system is an ordered pair (x, y) that makes both equations true at the same time.
Worked example 1
Is (3, 1) a solution of x + y = 4 and 2x − y = 5?
First equation: 3 + 1 = 4 ✓. Second equation: 2(3) − 1 = 5 ✓. Both are true, so (3, 1) is a solution. The pair (4, 0) satisfies the first equation but gives 2(4) − 0 = 8 in the second, so it is not a solution of the system.
The graph of a linear equation in two variables is a straight line. Every point on the line satisfies the equation. So a solution of the system is a point on both lines: their point of intersection.
Compare the ratios of the coefficients.
Worked example 2
Classify 3x − 6y = 9 and x − 2y = 5. The ratios are 3 ÷ 1 = 3, −6 ÷ ( − 2) = 3, and 9 ÷ 5 = 1.8. The first two are equal but the third is different, so the lines are parallel and there is no solution.
Tip
Exam technique. Always check a solution in both original equations: a pair that fits only one of them is the most common wrong answer offered in multiple-choice questions. To classify a system quickly, rewrite each equation as y = mx + c. Different slopes mean one solution; the same slope with different intercepts means none; the same slope and intercept means infinitely many.
Is (2, 5) a solution of y = 3x − 1 and x + y = 7?
Solution: 3(2) − 1 = 5 ✓ and 2 + 5 = 7 ✓. Yes.
Is (1, 4) a solution of 2x + y = 6 and x − y = 2?
Solution: 2 + 4 = 6 ✓ but 1 − 4 = −3, not 2. No: it fits only the first equation.
How many solutions has y = −x + 5 together with y = −x + 2?
Solution: Both slopes are −1 and the intercepts differ, so the lines are parallel: no solution.
How many solutions has x + 3y = 6 together with 2x + 6y = 12?
Solution: The ratios are 1 ÷ 2, 3 ÷ 6 and 6 ÷ 12, all equal to 0.5. It is the same line twice, so there are infinitely many solutions.
Remember
Sources and credits
The Rwandan worked examples, the practice items and the exercises were written by Ganzaa.
