Library / Foundations: Maths / FMA-45

Foundations: Maths

Two simultaneous equations: solution for y

Value of y that satisfies two simultaneous linear equations, found with Cramer’s rule.

Ref: FMA-45
Rev: v1
Field: Foundations: Maths
Result: y

y = \frac{a f - e c}{Δ} Δ = a d - b c (a x + b y = e, c x + d y = f)

With the same determinant Δ = ad − bc, the second unknown is y = (af − ec)/Δ. Solving for x and y together gives the single point where the two straight lines cross; the same pair of numerators are used in Cramer’s rule for larger systems.

Value of y that satisfies two simultaneous linear equations, found with Cramer’s rule. It is undefined when the determinant Δ is zero.

Calculator

Solve for

ySolution for yComputed

aCoefficient a of x (first equation)

−1010

bCoefficient b of y (first equation)

−1010

cCoefficient c of x (second equation)

−1010

dCoefficient d of y (second equation)

−1010

eRight-hand side of the first equation

−5050

fRight-hand side of the second equation

−5050

Determinant Δ = ad − bc−3
Solution for x2
Solution for y1
Check: a x + b y (should equal e)5
Check: c x + d y (should equal f)1

What drives the result

Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.

Solution for y () current value 1

fRight-hand side of the second equationKey driver

−31.735

aCoefficient a of x (first equation)

−1917

cCoefficient c of x (second equation)

−1323

eRight-hand side of the first equation

−17.316

dCoefficient d of y (second equation)

−93

bCoefficient b of y (first equation)

−4.54.5

For the same pair of equations as the x entry, y = 1 and x = 2. Both satisfy the two equations.
Changing a right-hand side moves one line parallel to itself, so the crossing point slides along the other line: the solution changes in proportion to the change, at a rate set by 1/Δ.

Chart data as tables

Ranked variables

Variable	Typical range	Lowest result	Highest result	Swing ()	Local step	Change in result
f Right-hand side of the second equation	−50 to 50	−31.67	35	66.67	+2	−133 %
a Coefficient a of x (first equation)	−10 to 10	−19	17	36	+0.4	−23.5 %
c Coefficient c of x (second equation)	−10 to 10	−13	23	36	+0.4	+47.1 %
e Right-hand side of the first equation	−50 to 50	−17.33	16	33.33	+2	+66.7 %
d Coefficient d of y (second equation)	−10 to 10	−9	3	12	+0.4	+36.4 %
b Coefficient b of y (first equation)	−10 to 10	−4.5	4.5	9	+0.4	−11.8 %

Sweep of Right-hand side of the second equation

Right-hand side of the second equation	Solution for y
−50	35
−37.5	26.67
−25	18.33
−12.5	10
0	1.667
12.5	−6.667
25	−15
37.5	−23.33
50	−31.67

Sweep one variable

Vary

Solution for y ()Hover, touch, or use the arrow keys on the chart

Variables and how each one contributes

Solution for y

Result ·

What the equation gives you. Everything else feeds it.

Coefficient a of x (first equation)

· typical −10 to 10

The coefficient of x in the first equation, ax + by = e. Changes y through the determinant and the right-hand sides; the size and sign of the effect depend on the other coefficients, so use the swept values to see it. Raising it by 1 (to 3) moves the result by −50 %, from 1 to 0.5.

Coefficient b of y (first equation)

· typical −10 to 10

The coefficient of y in the first equation. Changes y through the determinant and the right-hand sides; the size and sign of the effect depend on the other coefficients, so use the swept values to see it. Raising it by 1 (to 2) moves the result by −25 %, from 1 to 0.75.

Coefficient c of x (second equation)

· typical −10 to 10

The coefficient of x in the second equation, cx + dy = f. Changes y through the determinant and the right-hand sides; the size and sign of the effect depend on the other coefficients, so use the swept values to see it. Raising it by 1 (to 2) moves the result by +100 %, from 1 to 2.

Coefficient d of y (second equation)

· typical −10 to 10

The coefficient of y in the second equation. Changes y through the determinant and the right-hand sides; the size and sign of the effect depend on the other coefficients, so use the swept values to see it. Raising it by 1 (to 0) moves the result by +200 %, from 1 to 3.

Right-hand side of the first equation

· typical −50 to 50

The right-hand side of the first equation. Changes y through the determinant and the right-hand sides; the size and sign of the effect depend on the other coefficients, so use the swept values to see it. Raising it by 1 (to 6) moves the result by +33.3 %, from 1 to 1.333.

Right-hand side of the second equation

· typical −50 to 50

The right-hand side of the second equation. Changes y through the determinant and the right-hand sides; the size and sign of the effect depend on the other coefficients, so use the swept values to see it. Raising it by 1 (to 2) moves the result by −66.7 %, from 1 to 0.3333.

Worked example with the values above

Determinant: Δ = ad − bc = −3.
Numerator for y: af − ec = (2)(1) − (5)(1) = −3.
y = −3 / −3 = 1.

Assumptions and limits

The two equations are linear and consistent, with a nonzero determinant.
Exact arithmetic: if the determinant is very small compared with the coefficients, rounding in the inputs makes the answer very sensitive.

Common mistakes

Forgetting the minus sign in the determinant.
Using x’s numerator for y.

Statistics and uncertainty

As for x, errors in the coefficients are magnified by 1/Δ. Check the solution by substituting it into both equations, which should return e and f to the precision of the inputs.

Where it is used

Solving two unknown forces, flows or currents.
Finding where two straight lines intersect.

Maths you need

Determinant: ad − bc of a two-by-two array
Linear equation: one with no powers of the unknowns

Related formulas

Practise this formula