Systems of equations with two unknowns

What a solution is

A solution of a system is a pair $(x, y)$ that satisfies every equation at once. Each linear equation $ax + by = c$ is a straight line in the plane, so solving two of them means finding where the two lines meet.

Substitution

Isolate one unknown in the equation where it is easiest (a coefficient of $1$ or $-1$), and substitute into the other:

$$\begin{cases} x - y = 1 \\ 2x + 3y = 7 \end{cases} \quad\Rightarrow\quad x = 1 + y \quad\Rightarrow\quad 2(1 + y) + 3y = 7.$$

So $5y = 5$, $y = 1$ and $x = 2$. Substitution is the natural choice when one equation is not linear, for example a parabola and a line.

Elimination

Multiply the equations by numbers that make the coefficients of one unknown equal or opposite, then add or subtract the equations to remove it. For

$$\begin{cases} 3x + 2y = 16 \\ 5x - 2y = 8 \end{cases}$$

adding removes $y$ at once: $8x = 24$. Elimination is usually faster when both equations are linear and no coefficient is $1$.

Combinations: answer the question asked

A question may ask for an expression such as $x + y$ or $3x + 2y$, not for $x$ and $y$. Before solving, see whether adding or subtracting the equations produces that expression directly. In the system above, the sum of the two equations is $8x = 24$; in the first system, adding $x - y = 1$ to $2x + 3y = 7$ gives $3x + 2y = 8$ immediately.

One, none or infinitely many

Two lines in the plane meet once, never (parallel) or everywhere (the same line). For

$$\begin{cases} a_1 x + b_1 y = c_1 \\ a_2 x + b_2 y = c_2 \end{cases}$$

the test is the number $a_1 b_2 - a_2 b_1$:

Condition	Lines	Solutions
$a_1 b_2 - a_2 b_1 \ne 0$	cross	exactly one
$a_1 b_2 - a_2 b_1 = 0$, equations inconsistent	parallel, distinct	none
$a_1 b_2 - a_2 b_1 = 0$, one equation a multiple of the other	the same line	infinitely many

When $a_1 b_2 - a_2 b_1 = 0$, the left sides are proportional. Compare the right sides with the same factor: if they match, the equations say the same thing; if not, they contradict each other. For example, $x + 2y = 4$ and $3x + 6y = 12$ are the same line, while $x + 2y = 4$ and $3x + 6y = 5$ are parallel.

A line and a parabola

To intersect $y = x^2 + px + q$ with $y = mx + n$, substitute the line into the parabola. You get a quadratic in $x$, and its discriminant tells you how the line sits:

• $\Delta > 0$: two intersection points (a secant);
• $\Delta = 0$: one point, the line is tangent;
• $\Delta < 0$: no common point.

Each root $x$ gives a point; find its $y$ from the line, which is quicker, and check it in the parabola.

Word problems

Name the two unknowns explicitly, with units ("$a$ = number of adult tickets"), write one equation per independent fact, solve, then check the answer against the story: counts must be whole numbers, lengths positive. The check catches most setting-up errors.