Simultaneous Equations — Maths for General Chemistry

Two unknowns, two equations

Find the values that satisfy everything at once

To solve a system you need as many independent equations as unknowns. Two methods do the job for two unknowns:

Substitution

Rearrange one equation for a variable, then put it into the other.

Elimination

Add or subtract multiples of the equations to cancel a variable.

Graphically, each equation is a straight line and the solution is where they cross. If the lines are parallel (same gradient, different intercept) the system has no solution. If they are identical they share infinitely many solutions.

In thermochemistry, Hess's law uses the elimination method: you add or subtract known reactions (each with its $Δ H$ ) to reach a target reaction. In spectrophotometry you can measure two absorbances at two wavelengths and simultaneously solve for the concentrations of two species.

General system

a_{1} x + b_{1} y = c_{1}, a_{2} x + b_{2} y = c_{2}

Cramer's rule

x = \frac{c _{1} b _{2} - c _{2} b _{1}}{a _{1} b _{2} - a _{2} b _{1}}, y = \frac{a _{1} c _{2} - a _{2} c _{1}}{a _{1} b _{2} - a _{2} b _{1}}

A three-reaction Hess cycle is a system of three simultaneous equations in three unknowns: you match each species (CO₂, H₂O, etc.) to make them cancel except in the target equation. This is exactly Gaussian elimination applied to thermochemistry — each reaction is a row in the matrix, and each chemical species is a column.

In two-wavelength spectrophotometry, measuring absorbance at two wavelengths where the two components have different molar absorptivities gives two equations in the two unknown concentrations. Provided the absorptivities are sufficiently different at the two wavelengths (i.e. the determinant is large), the system is well-conditioned and the concentrations can be found accurately. A poorly designed wavelength pair (similar absorptivities, small determinant) gives an ill-conditioned system that amplifies measurement noise.

Charge and mass balances in solution chemistry are two independent constraints that together determine the two unknown concentrations in a system. For an aqueous NH₃ solution: mass balance $[N H_{3}] + [N H_{4}^{+}] = C$ and charge balance $[N H_{4}^{+}] + [H^{+}] = [O H^{-}]$ form a simultaneous pair.

Common pitfalls

Sign errors in elimination. When you multiply an equation by a constant before adding, remember to multiply every term including the right-hand side. A missing negative sign on one term produces a wrong result that is hard to spot.
Reversing a Hess equation changes the sign of ΔH. Reversing $A \to B$ to get $B \to A$ multiplies $Δ H$ by −1 (not +1).
Inconsistent or underdetermined systems. If two equations are multiples of each other (linearly dependent) there is no unique solution — infinitely many or none. Always check that the equations are genuinely independent (different ratios of coefficients).
Mixing up which species is in which row. In a spectrophotometry problem, the absorbance at 395 nm goes in one equation and at 510 nm in the other. Swapping them gives a wrong answer without an obvious error message.

In chemistry

Hess's-law cycles, mixtures of two absorbing species (two wavelengths, two equations), and mass balances all reduce to simultaneous equations. Open the two-line solver.

Two-line solver

Drag the coefficients to reposition the lines. The green dot marks their intersection — the simultaneous solution.

Adjust both lines — the dot is the simultaneous solution

a_{1} x + b_{1} y a_{2} x + b_{2} y = c_{1} = c_{2}

Line 1 (orange)

a₁1

b₁1

c₁5

Line 2 (blue)

a₂1

b₂-1

c₂1

solution (x, y)(3.00, 2.00)

Worked example 1Hess's law — elimination in action

Given the two reactions:

(1) $C (s) + O_{2} (g) \to C O_{2} (g), Δ H_{1} = - 393.5 kJ mo l^{- 1}$

(2) $CO (g) + \frac{1}{2} O_{2} (g) \to C O_{2} (g), Δ H_{2} = - 283.0 kJ mo l^{- 1}$

Find $Δ H$ for $C + \frac{1}{2} O_{2} \to CO$ .

Worked example 2Two-component mixture by simultaneous equations

A 250 mL solution contains a mixture of NaCl and KCl. The total mass of solute is 3.720 g and the molar concentration of Cl⁻ ions is 0.2400 mol dm⁻³. Find the mass of each salt. ( $M_{NaCl} = 58.44, M_{KCl} = 74.55 g mo l^{- 1}$ )

Worked examples

Two-wavelength spectroscopy and a thermochemical cycle — both reduce to simultaneous equations.

Worked example 3Beer–Lambert with two species

A solution of Co²⁺ and Ni²⁺ has absorbances $A_{395} = 0.810$ and $A_{510} = 0.480$ . Molar absorptivities (L mol⁻¹ cm⁻¹, l = 1 cm):

$ε_{Co, 395} = 36.4, ε_{Ni, 395} = 5.60$ and $ε_{Co, 510} = 5.60, ε_{Ni, 510} = 17.0$ .

Find $[C o^{2 +}]$ and $[N i^{2 +}]$ .

Worked example 4Finding formation enthalpies with Hess cycles

Three combustion enthalpies are known:

$Δ H_{comb}^{\circ} (C) = - 393.5 kJ mo l^{- 1}$

$Δ H_{comb}^{\circ} (H_{2}) = - 285.8 kJ mo l^{- 1}$

$Δ H_{comb}^{\circ} (C_{2} H_{5} OH) = - 1366.8 kJ mo l^{- 1}$

Find $Δ H_{f}^{\circ} (C_{2} H_{5} OH)$ .

Worked example 5Mass balance in a buffer — two equations for two unknowns

A buffer is made by mixing acetic acid and sodium acetate to give a total acetate concentration of 0.200 M and a pH of 4.44. Using $p K_{a} = 4.74$ and the Henderson–Hasselbalch equation as one constraint, plus the mass balance $[HA] + [A^{-}] = 0.200$ as the other, find $[HA]$ and $[A^{-}]$ .

Worked example 6Three-equation Hess cycle — combustion to formation enthalpy

Given three combustion enthalpies (at 298 K):

(1) $C (s) + O_{2} (g) \to C O_{2} (g), Δ H_{1} = - 393.5 kJ mo l^{- 1}$

(2) $H_{2} (g) + \frac{1}{2} O_{2} (g) \to H_{2} O (l), Δ H_{2} = - 285.8 kJ mo l^{- 1}$

(3) $C H_{3} OH (l) + \frac{3}{2} O_{2} (g) \to C O_{2} (g) + 2 H_{2} O (l), Δ H_{3} = - 726.0 kJ mo l^{- 1}$

Find $Δ H_{f}^{\circ} (C H_{3} OH)$ .

Worked example 7Two-wavelength Beer–Lambert (corrected — 2-component mixture)

A mixture of permanganate (MnO₄⁻) and chromate (CrO₄²⁻) gives absorbances: $A_{440} = 0.370$ and $A_{545} = 0.510$ (path length 1.00 cm). Molar absorptivities (L mol⁻¹ cm⁻¹):

$ε_{Mn, 440} = 95.5, ε_{Cr, 440} = 1840$

$ε_{Mn, 545} = 2350, ε_{Cr, 545} = 10.5$

Find $[Mn O_{4}^{-}]$ and $[Cr O_{4}^{2 -}]$ .

ChallengeChallenge — three-species spectrophotometric mixture

A solution contains Fe²⁺, Fe³⁺, and Cu²⁺. Absorbances at three wavelengths (path 1 cm):

$A_{340} = 0.600, A_{470} = 0.450, A_{590} = 0.300$

Molar absorptivities (L mol⁻¹ cm⁻¹):

$ε_{F e^{2 +}} : 500, 120, 30$ at 340, 470, 590 nm

$ε_{F e^{3 +}} : 2100, 850, 55$ at 340, 470, 590 nm

$ε_{C u^{2 +}} : 10, 30, 500$ at 340, 470, 590 nm

Set up the 3×3 system and solve for the three concentrations.

Check yourself

Four questions connecting simultaneous equations to chemical problem-solving.

Question 1 of 4 · Score 0

Geometrically, the solution of two simultaneous linear equations is:

Choose an answer.