Foundations — Linear equations in two variables — graphical and algebraic solutions

Before you can read a single line of the parent note, you need to own the symbols it throws at you. This page builds each one from nothing: plain words → a picture → why the topic can't live without it. Read top to bottom; each block leans on the one above.

Return to the parent any time: parent topic (Hinglish).

1. A variable — the letter that stands for "some number we don't know yet"

Think of $x$ as an empty box. Later we might discover the box must hold $2$. Until then it stays a letter so we can talk about it.

• Picture: a labelled box $\boxed{x}$ waiting to be filled.
• Why the topic needs it: the whole point is to find the numbers hiding inside $x$ and $y$. No letters, nothing to solve for.

2. The coordinate plane — a map made of two number lines

To draw a rule about two numbers at once, we need a place to draw. That place is the coordinate plane.

• Picture: two rulers glued at right angles — see the figure above. Right/up are positive; left/down are negative.
• Why the topic needs it: an equation in two variables becomes a shape only once we have a place to plot every $(x,y)$ pair.

3. An ordered pair $(x,y)$ — an address on the map

• Picture: the figure shows how $(3,2)$ is reached — right $3$, up $2$ — landing on one dot.
• Why the topic needs it: every solution of a system is one such pair. When the parent says "find $(x,y)$", it means "find this one address."

4. The equal sign and what "solving" really means

Example: for $2x+3y=12$, the pair $(3,2)$ is a solution because $2(3)+3(2)=6+6=12$ — the two sides genuinely match.

• Picture: a balance scale, both pans level. A solution keeps it balanced.
• Why the topic needs it: "solving" is nothing more than finding the pairs that keep the scale level.

5. Coefficients and constants — the fixed numbers in $ax+by+c$

• Picture: $a$ and $b$ are dials set to a value; $x,y$ are the sliders you move.
• Why the topic needs it: the entire consistency test compares the coefficients $a_1,b_1,c_1$ of one line with $a_2,b_2,c_2$ of another. You must know which number is which.

6. Degree 1 — why the graph is straight and not curved

• Picture: the figure contrasts a straight line (degree $1$) with a curved parabola (degree $2$). Degree $1$ ⇒ no bending.
• Why the topic needs it: "linear" means straight. The parent's promise that every equation is a straight line depends entirely on this degree-$1$ rule. One $x^2$ and the whole method collapses.

7. Slope $m$ — the steepness that makes a line a line

The parent writes $y=mx+k$ and says "constant slope." Here is what slope is, from zero.

• Picture: the figure shows a right-triangle "step" under the line: run across, rise up. Same triangle fits anywhere along the line — that's what constant slope means.
• Why the topic needs it: two lines are parallel exactly when their slopes are equal. That single fact powers the "no solution" and "same line" cases in the parent.

8. Ratios — the fractions that compare two lines

The consistency test compares $\frac{a_1}{a_2}$, $\frac{b_1}{b_2}$, $\frac{c_1}{c_2}$. Make sure these are second nature.

• Picture: two recipes; if every ingredient in recipe 2 is exactly double recipe 1, the ratios all match and the recipes taste identical (same line!).
• Why the topic needs it: when all three ratios match, one equation is just a scaled copy of the other — same line, infinite solutions. When only the first two match, same slope but shifted — parallel, no solution.

9. Subscripts $a_1,a_2$ — labels, not maths

• Picture: two envelopes labelled "Line 1" and "Line 2"; inside each are its own $a$, $b$, $c$.
• Why the topic needs it: a system has two equations, so we need two sets of coefficients without confusing them. Subscripts keep the bookkeeping clean.

Prerequisite map

Equipment checklist

Test yourself — cover the right side and answer aloud.