Two equations with two unknowns
Two equations, two unknowns: the solution is the point where the two lines cross. Enter the coefficients and see both the working and the graph.
The explanation is shown. Quick: six numbers in, x and y out. Advanced: each equation as a line y = ax + b, and what the solution becomes if c₁ changes.
Your numbers
a₁x + b₁y = c₁
a₂x + b₂y = c₂
1x + 1y = 10
1x − 1y = 2
Answer
x
6
y
4
How to work it out
- 1D = a₁·b₂ − a₂·b₁ = 1·-1 − 1·1 = -2
- 2Dx = c₁·b₂ − c₂·b₁ = 10·-1 − 2·1 = -12
- 3Dy = a₁·c₂ − a₂·c₁ = 1·2 − 1·10 = -8
- 4x = Dx / D = 6 · y = Dy / D = 4
- 5Test: 10 = 10 · 2 = 2
The formula
x = Dx / D · y = Dy / D
The determinant method. D is a₁·b₂ − a₂·b₁; if it is 0, the lines are parallel.
Elimination method
Add the equations or subtract one from the other, so that one unknown disappears.
The copper line is the first equation, the dashed one the second. The solution is where they cross.
In depth: more numbers and “what if”
The equations as lines
| Equation | As a line | Slope |
|---|---|---|
| (1) 1x + 1y = 10 | y = -1x + 10 | -1 |
| (2) 1x − 1y = 2 | y = 1x − 2 | 1 |
What if c₁ changes?
| c₁ | x | y |
|---|---|---|
| 8 | 5 | 3 |
| 9 | 5.5 | 3.5 |
| 10 | 6 | 4 |
| 11 | 6.5 | 4.5 |
| 12 | 7 | 5 |
More: Equations and functions
- EquationsSolve equations and inequalities step by step — and isolate a quantity in a formula.
- Formula with three quantitiesThe formula triangle: find the third quantity in A = B · C — e.g. U = R · I.
- FunctionsQuadratic function, inverse proportionality and power function — with a graph.
- Linear functiony = ax + b as a graph and table — the line through two points or the line of best fit.
- Scale and ratiosFrom drawing to reality, area and volume to scale, the rule of three and dividing in a ratio.
- Growth and decayPercentage growth, the growth rate from two measurements, and doubling time.