Foundations — Solving 2×2 systems using Cramer's rule

This page assumes you have seen nothing. We build every symbol the parent note (Cramer's rule) throws at you, in an order where each new idea only leans on the ones before it.

1. An unknown, and the letters $x$ and $y$

Picture it. Think of $x$ as an empty box $\square$. The equation is a clue about what number belongs in the box. When we "solve," we are hunting for the one number that makes every clue true.

Why the topic needs it. Cramer's rule exists to find two unknowns at once — the two secret numbers hidden inside two equations. Without the idea of "a letter standing for a number we must discover," there is nothing to solve.

2. A linear equation, and what "linear" means

Picture it — this is the most important picture on the page. Every linear equation in $x$ and $y$ draws a perfectly straight line when you mark all the $(x,y)$ points that satisfy it. That is why it is called linear — its graph is a line.

Look at the figure: pick any point on the red line, read off its horizontal position ($x$) and vertical position ($y$), plug them into $x+2y=4$, and it works out. Points off the line fail the equation.

Why the topic needs it. The whole parent note is about two such equations. Each one is a line. The answer is where they meet — so we must first agree that one equation = one line.

3. The coefficient — the number multiplying an unknown

Reading the subscripts. The little number below — the subscript — is just a name tag, not a multiplication. $a_1$ means "the $a$ from equation 1"; $a_2$ means "the $a$ from equation 2." They are two different fixed numbers that happen to play the same role (both are "the coefficient of $x$") in two different equations.

Why the topic needs it. Cramer's rule is built entirely out of these coefficients — arranged in a grid. If you can't tell $a_1$ (coefficient of $x$, row 1) from $b_1$ (coefficient of $y$, row 1), the grid is meaningless.

4. The constant $c$ — the answer on the right

Picture it. In the line figure, $c$ controls how far the line sits from the origin. Change $c$ and the line slides across the page, keeping the same tilt.

Why the topic needs it. The two constants $c_1,c_2$ form the "answer column." Cramer's magic is to slide this column into the coefficient grid — so we need to know it as a thing on its own.

5. The system — two equations, both true at once

Picture it. Two lines on one sheet of paper.

The solution is the single point (red dot) lying on both lines at once — the only $(x,y)$ that obeys both clues.

Why the topic needs it. This crossing point is the answer Cramer's rule computes. Everything downstream is about locating this dot.

6. The matrix and the coefficient matrix

Picture it — burn this into memory.

Notice the red column: the column owns a variable. Column 1 belongs to $x$, column 2 belongs to $y$. This single fact is why Cramer replaces a column (never a row).

Why the topic needs it. The determinant we're about to meet chews on this grid. And the "column swap" that finds each unknown only makes sense once you see that $x$ lives in column 1.

See also Matrix form of linear equations Ax=b, which packs the whole system into $A\mathbf{x}=\mathbf{c}$.

7. The determinant — the criss-cross number

Now the star symbol of the parent note.

Why this exact recipe — "main diagonal minus anti-diagonal"? Multiply the two numbers going down-right (the main diagonal, $a$ and $d$), then subtract the two going down-left (the anti-diagonal, $b$ and $c$). The tool answers a specific question: "Do the two rows of this grid point in genuinely different directions, or are they secretly parallel?" When the answer is $0$, they're parallel.

Picture it — what the number measures. Treat each row as an arrow from the origin: $(a,b)$ and $(c,d)$. Those two arrows span a parallelogram. The determinant is the (signed) area of that parallelogram, shown in red. If the arrows line up, the parallelogram squashes flat — zero area — and $\det =0$.

Every sign case (so you're never surprised):

• $ad-bc>0$: rows spread "counter-clockwise" — positive area.
• $ad-bc<0$: rows spread "clockwise" — the signed area is negative. Perfectly normal; Cramer divides two such signed numbers and the signs sort themselves out (Example 1 in the parent divides $-26$ by $-14$).
• $ad-bc=0$: rows parallel — degenerate, area collapses.

Why the topic needs it. Cramer's rule is nothing but three of these determinants divided into ratios. Full treatment lives at Determinant of a 2×2 matrix.

8. Division, and why "$D\ne 0$" is the whole ballgame

Cramer's rule writes $x=\frac{D_x}{D}$. The bottom is $D=\det A$. From section 7, $D=0$ exactly when the two lines are parallel — no clean crossing point — so it makes perfect sense that the formula refuses to give an answer there. The algebra (undefined division) and the geometry (parallel lines) tell the same story.

The three outcomes — one solution, none, or infinitely many — are sorted by Consistency of linear systems.

9. Putting the notation together

You now own every symbol the parent uses:

Prerequisite map

Equipment checklist