What is a pair of linear equations?
Two linear equations in the same two variables, such as
x + 2y = 8 and 3x − y = 3,
form a pair of linear equations. A solution is a pair (x, y) that satisfies both equations at once. Here, (2, 3) works: 2 + 6 = 8 and 6 − 3 = 3.
The graphical method
Each equation is a straight line. Two lines in a plane can:
Lines | Number of solutions | The pair is |
|---|---|---|
intersect at one point | exactly one (the point of intersection) | consistent |
be parallel | none | inconsistent |
coincide (be the same line) | infinitely many | consistent (dependent) |
Deciding from the coefficients
For a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0:
Condition | Lines | Solutions |
|---|---|---|
a₁/a₂ ≠ b₁/b₂ | intersecting | exactly one |
a₁/a₂ = b₁/b₂ ≠ c₁/c₂ | parallel | none |
a₁/a₂ = b₁/b₂ = c₁/c₂ | coincident | infinitely many |
Example. 2x + 3y = 9 and 4x + 6y = 18. Ratios: 2/4 = 3/6 = 9/18 = ½. Coincident lines: infinitely many solutions.
Example. x − 2y = 4 and 2x − 4y = 5. 1/2 = −2/−4 = ½ but 4/5 ≠ ½. Parallel lines: no solution.
Example. For what k does kx + 3y = 5, 6x + 9y = 7 have no solution? We need k/6 = 3/9 ≠ 5/7, so k = 2.
The substitution method
From one equation, express one variable in terms of the other.
Substitute into the other equation and solve.
Substitute back to find the second variable.
Example. x + y = 14 and x − y = 4.
From the first, x = 14 − y. Substitute: 14 − y − y = 4, so 2y = 10 and y = 5. Then x = 9.
The elimination method
Multiply the equations by suitable numbers so that one variable has the same coefficient (or opposite coefficients) in both.
Add or subtract to eliminate it.
Solve for the remaining variable and substitute back.
Example. 3x + 4y = 10 and 2x − 2y = 2.
Multiply the second by 2: 4x − 4y = 4. Add to the first: 7x = 14, so x = 2. Then 3(2) + 4y = 10 gives y = 1.
If eliminating one variable also eliminates the other:
you get a true statement like 0 = 0: infinitely many solutions;
you get a false statement like 0 = 3: no solution.
Word problems
Example: ages. Six years ago, Meera was four times as old as her brother Kabir. Four years from now, she will be twice as old as he will be. Let their present ages be x and y.
x − 6 = 4(y − 6), which gives x − 4y = −18.
x + 4 = 2(y + 4), which gives x − 2y = 4.
Subtracting the first from the second: 2y = 22, so y = 11, and then x = 26. Meera is 26 and Kabir is 11.
Example: fractions. A fraction becomes 2/3 if 1 is added to both its numerator and denominator, and 4/7 if 1 is subtracted from both. Let it be x/y.
3(x + 1) = 2(y + 1), which gives 3x − 2y = −1.
7(x − 1) = 4(y − 1), which gives 7x − 4y = 3.
Doubling the first gives 6x − 4y = −2; subtracting it from the second gives x = 5, and then y = 8. The fraction is 5/8.
Example: speed. A boat goes 24 km upstream and 60 km downstream in 8 hours, and 40 km upstream and 48 km downstream in 9 hours. Let the boat's speed in still water be u and the stream's v. Write a = 1/(u − v) and b = 1/(u + v). Then 24a + 60b = 8 and 40a + 48b = 9, which give a = 1/8 and b = 1/12. So u − v = 8 and u + v = 12, giving u = 10 km/h and v = 2 km/h.
Summary
A pair of linear equations has one, no or infinitely many solutions, shown by intersecting, parallel or coincident lines.
Compare a₁/a₂, b₁/b₂, c₁/c₂ to decide which case holds.
Solve algebraically by substitution or elimination; check by substituting into both equations.