Reference · module 5
Two equations at once
Everything this module explains, on one page.
Substitution
Solve a pair of equations by putting one into the other.
- Two unknowns need two equations. x + y = 10 alone has endless answers, but together with y = x + 2 only one pair fits both: x = 4, y = 6.
- Substitution: if one equation says y = x + 2, replace y in the other with x + 2. Then x + (x + 2) = 10 has one unknown: 2x = 8, x = 4.
- Then put x back to find y: y = 4 + 2 = 6. Check both equations: 4 + 6 is 10 and 6 is 4 + 2. A solution must satisfy both.
The mistake you are about to make
x = 4 — done
x = 4, y = 6
Finding x feels like the end of the work. A system asks for both unknowns, so the answer is incomplete until y is found too.
Elimination
Solve systems by adding or subtracting the equations.
- Equations can be added side by side. In x + y = 9 and x − y = 3, adding them makes y cancel: 2x = 12, so x = 6 and y = 3.
- If an unknown has the same coefficient in both, subtract instead. 3x + 2y = 16 minus x + 2y = 8 gives 2x = 8, so x = 4.
- Sometimes multiply one equation first so coefficients match. For x + y = 5 and 2x + 3y = 12, double the first: 2x + 2y = 10, then subtract: y = 2.
The mistake you are about to make
(x + y) + (x − y) = 2x + 2y
(x + y) + (x − y) = 2x
Adding looks like it should keep every letter. The +y and −y cancel each other, which is exactly why the equations were added.
Systems in word problems
Set up two equations from a story and solve them.
- Many stories have two unknowns: the price of an apple and of a pear, the number of cars and of bikes. Name each with a letter before writing anything.
- Each fact in the story gives one equation. 10 vehicles: c + b = 10. 32 wheels: 4c + 2b = 32. Two facts, two equations.
- Solve with either method, then check the answer against the story, not just the equations. 6 cars and 4 bikes: 10 vehicles and 24 + 8 = 32 wheels.
The mistake you are about to make
c + b = 32
4c + 2b = 32
32 is a total, like 10, so it seems to count vehicles too. It counts wheels, and each car brings four of them.