Systems of Linear Equations

A fruit seller buys apples and oranges. 3 kg of apples and 2 kg of oranges cost ₹260. 2 kg of apples and 3 kg of oranges cost ₹240. What is the price per kilogram of each fruit?

Let the price of apples be x ₹/kg and oranges be y ₹/kg. The two conditions give two equations:

Each equation alone describes a line in the xy-plane — infinitely many pairs (x, y) satisfy it. But you need a pair that satisfies both equations at once. That pair is the single point where the two lines cross. Finding that crossing point is the central problem of this article.

The graphical method

The most direct approach: plot both lines and see where they meet.

For 3x + 2y = 260:

• x-intercept: y = 0 \implies x = 260/3 \approx 86.7.
• y-intercept: x = 0 \implies y = 130.

For 2x + 3y = 240:

• x-intercept: y = 0 \implies x = 120.
• y-intercept: x = 0 \implies y = 80.

Plot both lines. They cross at one point. Reading the coordinates carefully gives x = 60, y = 40. Check: 3(60) + 2(40) = 180 + 80 = 260. 2(60) + 3(40) = 120 + 120 = 240. Both equations satisfied.

The graphical method is valuable for seeing the answer, but it has a practical limitation: if the solution involves fractions or large numbers, reading exact coordinates from a graph is unreliable. For precise answers, you need an algebraic method.

The substitution method

Express one variable in terms of the other from one equation, then substitute into the second.

Take the same system:

From equation (1): 2y = 260 - 3x, so y = \frac{260 - 3x}{2}.

Substitute into equation (2):

Multiply both sides by 2 to clear the fraction:

Now back-substitute: y = \frac{260 - 3(60)}{2} = \frac{260 - 180}{2} = \frac{80}{2} = 40.

The solution is (60, 40), the same point as before. Substitution works well when one equation is easy to solve for one variable — especially when a coefficient is 1.

The elimination method

Instead of substituting, multiply the equations by chosen constants so that one variable has the same coefficient in both, then subtract to eliminate it.

Take the same system:

Multiply (1) by 3 and (2) by 2:

Subtract (2') from (1'):

Substitute x = 60 into equation (2): 2(60) + 3y = 240 \implies 120 + 3y = 240 \implies 3y = 120 \implies y = 40.

The name "elimination" says it all — you manufacture equal coefficients, then subtract to eliminate one variable. The arithmetic is usually cleaner than substitution because you avoid fractions.

The cross-multiplication method

For the general system

there is a direct formula. Write the coefficients in two rows:

provided a_1 b_2 - a_2 b_1 \neq 0. This gives:

For the fruit-price problem, rewrite the equations as 3x + 2y - 260 = 0 and 2x + 3y - 240 = 0. Here a_1 = 3, b_1 = 2, c_1 = -260, a_2 = 2, b_2 = 3, c_2 = -240.

Same answer: (60, 40). The cross-multiplication method is fastest when you want a direct formula answer — no intermediate steps, no back-substitution. The quantity a_1 b_2 - a_2 b_1 in the denominator is called the determinant of the system and appears again in Determinants.

A memory aid for the cross-multiplication formula: write the coefficients a, b, c in a cycle, and cross-multiply adjacent pairs with alternating signs. The pattern is

Cross-multiply diagonally: b_1 c_2 - b_2 c_1 gives the numerator for x, c_1 a_2 - c_2 a_1 gives the numerator for y, and a_1 b_2 - a_2 b_1 gives the common denominator.

Consistency and types of solutions

Not every system has a single solution. The geometry makes this clear: two lines in a plane can relate to each other in exactly three ways.

Case 1: Intersecting lines. The lines cross at one point. The system has exactly one solution — it is consistent and independent.

Case 2: Parallel lines. The lines never meet. The system has no solution — it is inconsistent.

Case 3: Coincident lines. The two equations describe the same line. The system has infinitely many solutions — it is consistent and dependent.

You can tell which case you are in from the coefficients alone, without solving. For the system a_1 x + b_1 y + c_1 = 0 and a_2 x + b_2 y + c_2 = 0:

The logic: if a_1/a_2 = b_1/b_2, the two lines have the same slope, so they are parallel. If they also have the same intercept (c_1/c_2 matches), they are the same line. If the slopes differ (a_1/a_2 \neq b_1/b_2), the lines must cross somewhere.

Example of no solution. Take 2x + 4y = 10 and x + 2y = 8. Rewrite the first as x + 2y = 5. Now the system is x + 2y = 5 and x + 2y = 8 — the same left side, different right side. The lines are parallel (a_1/a_2 = b_1/b_2 = 1, but c_1/c_2 = 5/8 \neq 1). No point can make x + 2y equal to both 5 and 8 at once.

Example of infinitely many solutions. Take 2x + 4y = 10 and x + 2y = 5. The first equation is just twice the second. They describe the same line. Every point on x + 2y = 5 is a solution: (5, 0), (1, 2), (-3, 4), and infinitely many others.

Interactive exploration

Drag the red point below along the line x + y = 7. The readout shows the values of x and y, and also computes 2x + y (the left side of the second equation 2x + y = 10). When 2x + y hits 10, you have found the intersection point — the solution that satisfies both equations simultaneously.

Two worked examples

Common confusions

• "Two equations in two unknowns always have a solution." Not if the equations are inconsistent (parallel lines). The system x + y = 5 and x + y = 8 has no solution.

• "If I get 0 = 0 while solving, I made an error." Not necessarily. Getting 0 = 0 means the two equations are the same line (dependent system). There are infinitely many solutions, and the algebra correctly collapses to a trivially true statement.

• "I need to check my answer in only one equation." Check in both equations. A correct solution satisfies both. An arithmetic mistake in one equation might produce a point that satisfies the first equation but not the second.

• "Cross-multiplication works even when a_1 b_2 - a_2 b_1 = 0." No — if the determinant is zero, the formula divides by zero. A zero determinant means the lines are parallel (or coincident), and there is no unique solution to find.

• "Elimination means I can only subtract." You can add too. If one equation has +3y and the other has -3y, adding the equations eliminates y. The choice between adding and subtracting depends on the signs of the matched coefficients.

• "The graphical method and algebraic methods give different answers." They never should. If they seem to, either the graph was read imprecisely (fractional coordinates are hard to read) or the algebra has an error. The geometric picture and the algebraic computation describe the same system.

Going deeper

The determinant

The quantity a_1 b_2 - a_2 b_1 that appears in the cross-multiplication formula is a 2 \times 2 determinant:

When the determinant is nonzero, the system has a unique solution. When it is zero, the system is either inconsistent or dependent. The determinant is a single number that encodes whether the two lines are "independent enough" to pin down a point. You meet determinants in full generality in Determinants, where they extend to 3 \times 3 and larger systems.

Cramer's rule

The cross-multiplication formula is a special case of Cramer's rule, which works for any system of n equations in n unknowns (provided the main determinant is nonzero). For a 2 \times 2 system a_1 x + b_1 y = d_1 and a_2 x + b_2 y = d_2:

The rule replaces one column of the coefficient matrix with the constants from the right-hand side, computes the determinant, and divides by the main determinant. The pattern extends to 3 \times 3 systems with three equations and three unknowns.

Systems of three or more equations

With three unknowns, you need three independent equations. Geometrically, each equation in three variables describes a plane in \mathbb{R}^3. Two planes generally meet in a line, and three planes generally meet at a point. The methods extend: elimination works by reducing a 3 \times 3 system to a 2 \times 2 system (eliminating one variable), then to a 1 \times 1 system. This cascading elimination is formalised as Gaussian elimination, which is the workhorse algorithm for solving linear systems of any size. You meet it in Matrices — Introduction.

Why systems matter

Almost every real-world problem with multiple interacting quantities reduces to a system of equations. Circuit analysis (Kirchhoff's laws) gives a system of linear equations. Balancing chemical equations is a system. Fitting a line to data (least squares) involves solving a system. Economic equilibrium models are systems. The 2 \times 2 case you learned here is the smallest nontrivial version — the training ground for a technique that scales to thousands of equations solved by computers in microseconds.

Where this leads next

Systems of linear equations connect algebra to geometry, to matrices, and eventually to almost every applied field.

• Linear Equations in Two Variables — the building blocks: each equation in the system is a line on the plane.
• Linear Equations in One Variable — what a system reduces to after elimination: a single equation in one unknown.
• Matrices — Introduction — the language that organises systems with three or more equations into compact notation.
• Determinants — the single number that decides whether a system has a unique solution.
• Coordinate Geometry Basics — where lines, distances, and intersection points live in the plane.