Linear equations in two unknowns

A system of linear equations in two unknowns is a pair of equations in variables $x$ and $y$, where each equation is linear (the highest power of each variable is 1). A solution is an ordered pair $(x, y)$ that satisfies both equations at the same time.

Assumed knowledge

Expand and collect like terms, e.g. $2(x+3) = 2x + 6$.
Substitute a value into an expression, e.g. replace $x$ by $4$ in $3x - 1$.
Add or subtract equations term by term when both sides are equal.

Standard form

A convenient form is \[ ax + by = c, \qquad dx + ey = f, \] where $a, b, c, d, e, f$ are constants. The same system can be written in other equivalent forms — for example $y = 2x + 1$ is the same as $-2x + y = 1$.

Substitution

Substitution means isolate one unknown in one equation, then replace that expression in the other equation.

Example. Solve

\[ \begin{cases} y = 2x + 1 \\ x + y = 7 \end{cases} \]

From the first equation, $y = 2x + 1$. Substitute into the second: \[ x + (2x + 1) = 7 \Rightarrow 3x + 1 = 7 \Rightarrow 3x = 6 \Rightarrow x = 2. \] Then $y = 2(2) + 1 = 5$. The solution is $(2, 5)$.

Check: $2(2)+1 = 5$ and $2 + 5 = 7$.

Elimination

Elimination adds or subtracts the equations so that one unknown cancels. You may need to multiply one or both equations first so that the coefficients of $x$ or $y$ match in size (one positive, one negative).

Example. Solve

\[ \begin{cases} x + y = 5 \\ 2x - y = 1 \end{cases} \]

The coefficients of $y$ are $1$ and $-1$, so add the equations: \[ (x+y) + (2x-y) = 5 + 1 \Rightarrow 3x = 6 \Rightarrow x = 2. \] Substitute into $x + y = 5$: $2 + y = 5$, so $y = 3$. Solution: $(2, 3)$.

Graphical interpretation

Each equation $ax + by = c$ (with $b \neq 0$) can be rearranged to $y = -\frac{a}{b}x + \frac{c}{b}$, which is a straight line. The solution of the system is the point where the two lines meet — their intersection.

Lines	Algebraic picture	Solutions
Intersect at one point	Different slopes	Exactly one $(x, y)$
Parallel, distinct	Same slope, different intercepts	No solution
Coincident (same line)	One equation is a multiple of the other	Infinitely many

Try it — graph explorer

Adjust the two lines or use presets. Read off the intersection and how the classification changes when the lines are parallel or coincident.

Try it — step-by-step solver

Pick a preset system and watch substitution or elimination steps. Systems with no unique solution show an explanation instead of incorrect steps.

Word problems

For story problems, define unknowns clearly, write two independent equations, then solve by substitution, elimination, or a graph.

Example (tickets). Adult tickets cost \$12 and child tickets \$7. A group buys 10 tickets for \$95. How many of each type?

Let $a$ be the number of adult tickets and $c$ the number of child tickets. Then $a + c = 10$ and $12a + 7c = 95$. From the first, $c = 10 - a$. Substitute: \[ 12a + 7(10 - a) = 95 \Rightarrow 12a + 70 - 7a = 95 \Rightarrow 5a = 25 \Rightarrow a = 5, \] so $c = 5$. Five adult and five child tickets.

History

Problems that need two unknowns at once are ancient. The Chinese text Nine Chapters on the Mathematical Art (around 200 BCE, with later commentary) includes fangcheng methods — arranging coefficients in a table and eliminating unknowns, much like modern elimination.

In the 9th century, al-Khwarizmi organised linear and quadratic problems in systematic rules that spread to Europe. By the 17th century, Descartes linked algebra with geometry: an equation in $x$ and $y$ describes a curve, so solving a system is finding where two curves meet. Today, computers solve huge systems, but the same ideas — replace, combine, or intersect — remain the core of school algebra.

Derivation

Why elimination preserves solutions

If $(x_0, y_0)$ satisfies both $ax + by = c$ and $dx + ey = f$, then any linear combination of the two equations is still true at $(x_0, y_0)$. For example, adding the left sides and adding the right sides gives \[ (a+d)x + (b+e)y = c + f, \] which $(x_0, y_0)$ also satisfies. Elimination chooses combinations so one unknown disappears, leaving a single equation in one unknown. Multiplying an equation by a non-zero constant before adding does not change the solution set either.

Why the graph solution is the intersection

The graph of $ax + by = c$ is all points $(x, y)$ that make the equation true. A solution of the system must lie on both graphs, so it must be a point common to both lines. If the lines cross once, that point is the unique solution; if they never meet, there is no solution; if they are the same line, every point on the line works.

Checkpoints

Pause and reason before continuing. Discuss with a partner or write your work in a notebook.

Checkpoint

In the graph explorer, use the “Parallel” preset. What does the status panel say about solutions? Does the canvas show an intersection point?

Checkpoint

Solve $y = 3x - 2$ and $2x + y = 8$ by substitution without a calculator. Does your pair $(x, y)$ satisfy both originals?

Checkpoint

For $x + 2y = 8$ and $2x + 4y = 16$, are the two equations independent, or is one a multiple of the other? How many solutions do you expect?

Checkpoint

When eliminating, why do we sometimes subtract equations instead of add them? Give an example where subtracting cancels $x$.

Checkpoint

A shop sells notebooks for \$4 and pens for \$2. You buy 6 items for \$18. What two equations describe this? Which method would you use first?

Checkpoint

In the step-by-step workbench, run elimination on the ticket-style preset. At which step does $y$ (or your chosen variable) disappear?

Applications

Ticket pricing

Cinemas and transport systems often use different prices for adults and children. If you know the total number of tickets and the total cost, you have two equations in two unknowns — the counts of each type. The same structure appears in fundraising dinners (plate vs concession) and sports seating.

Mixtures and concentrations

A chemist may mix a $20\%$ solution with a $50\%$ solution to obtain 2 L of $35\%$ solution. Let $x$ and $y$ be the volumes (in litres) of the weaker and stronger solutions. Then $x + y = 2$ (total volume) and $0.2x + 0.5y = 0.35 \times 2$ (total amount of pure solute). Solving the system gives the recipe.

Two constraints in business

A small business might need at least 100 units to cover fixed costs and also stay within a production cap of 150 units while hitting a revenue target. Each constraint is linear; the feasible plan is where the constraints overlap — a preview of ideas used later in linear programming.

Question bank

Work each question, then open the solution to check your reasoning. Difficulty increases through the sets.

Easy

Solve by substitution: $y = x + 2$ and $x + y = 8$.

Answer: $(3, 5)$.

Steps: Substitute $y = x + 2$ into $x + y = 8$: $x + (x+2) = 8$, so $2x = 6$, $x = 3$, $y = 5$.

Solve by elimination: $x + y = 6$ and $x - y = 2$.

Answer: $(4, 2)$.

Steps: Add: $2x = 8$, $x = 4$. Then $y = 6 - 4 = 2$.

Is $(1, 2)$ a solution of $2x + y = 4$ and $x - y = -1$?

Answer: Yes.

Check: $2(1)+2 = 4$ and $1 - 2 = -1$.

Intermediate

Solve: $2x + y = 7$ and $x - y = 2$.

Answer: $(3, 1)$.

Steps: Add equations: $3x = 9$, $x = 3$. Then $3 - y = 2$, $y = 1$.

Solve: $3x + 2y = 12$ and $x + y = 5$.

Answer: $(2, 3)$.

Steps: From $x + y = 5$, $y = 5 - x$. Substitute: $3x + 2(5-x) = 12$, $3x + 10 - 2x = 12$, $x = 2$, $y = 3$.

Two lines have equations $y = 2x + 1$ and $y = 2x - 3$. How many solutions does the system have?

Answer: No solution (parallel lines).

Explanation: Same slope $2$, different intercepts — the lines never meet.

Difficult

Solve: $2x + 3y = 13$ and $4x - y = 5$.

Answer: $(2, 3)$.

Steps: From $4x - y = 5$, $y = 4x - 5$. Substitute into $2x + 3y = 13$: $2x + 3(4x-5) = 13$, $2x + 12x - 15 = 13$, $14x = 28$, $x = 2$, $y = 3$.

Eliminate $x$: $3x + 2y = 11$ and $5x - 2y = 13$.

Answer: $(3, 1)$.

Steps: Add equations: $8x = 24$, $x = 3$. Then $3(3) + 2y = 11$, $2y = 2$, $y = 1$.

A purse has 12 coins, all 5¢ or 10¢, with total value 95¢. How many of each?

Answer: 5 coins at 5¢ and 7 coins at 10¢.

Steps: Let $n$ be 5¢ coins and $m$ be 10¢ coins. Then $n + m = 12$ and $5n + 10m = 95$. From the first, $n = 12 - m$. Substitute: $5(12-m) + 10m = 95$, $60 - 5m + 10m = 95$, $5m = 35$, $m = 7$, $n = 5$.

Hardcore

For what value of $k$ does $x + ky = 4$ and $2x + 2ky = 8$ have infinitely many solutions?

Answer: Every $k$ (the second equation is $2\times$ the first).

Explanation: The lines are coincident for all $k$ where the second is a multiple of the first; here they are always multiples, so infinitely many solutions.

Solve: $\dfrac{x}{2} + y = 5$ and $x - y = 1$.

Answer: $(4, 3)$.

Steps: Multiply the first equation by 2: $x + 2y = 10$. With $x - y = 1$, substitute $x = 1 + y$ into $x + 2y = 10$: $1 + y + 2y = 10$, $3y = 9$, $y = 3$, $x = 4$.

A rectangle has perimeter 26 cm. The length is 4 cm more than the width. Find the dimensions.

Answer: width $4.5$ cm, length $8.5$ cm.

Steps: $2\ell + 2w = 26$, $\ell = w + 4$. Substitute: $2(w+4)+2w=26$, $4w+8=26$, $w=4.5$, $\ell=8.5$.

References

The Nine Chapters on the Mathematical Art — early elimination methods (fangcheng).
al-Khwarizmi, Al-Kitāb al-mukhtaṣar fī ḥisāb al-jabr wal-muqābala (c. 820 CE).
René Descartes, La Géométrie (1637) — coordinates and curves.
HKDSE Compulsory Part curriculum — Number and Algebra strand (equations in two unknowns).

Last modified: 2026-05-22 23:34 UTC

Lines	Algebraic picture	Solutions
Intersect at one point	Different slopes	Exactly one \((x, y)\)
Parallel, distinct	Same slope, different intercepts	No solution
Coincident (same line)	One equation is a multiple of the other	Infinitely many