Linear Equations in Two Variables

1. Standard form — $ax + by + c = 0$

Any equation of the form $ax + by + c = 0$ is called a linear equation in two variables, where:

• $a$, $b$, $c$ are real numbers (can be integers, fractions, decimals, or zero).
• $a$ and $b$ are called coefficients of $x$ and $y$ respectively.
• $c$ is the constant term.
• Crucial restriction: $a$ and $b$ cannot both be zero simultaneously — otherwise the "equation" would collapse to $c = 0$, which has no variables at all.
• The word "linear" means the highest power of each variable is exactly 1.

Examples of linear equations in two variables:

• $2x + 3y = 7 \;\Rightarrow\; 2x + 3y - 7 = 0$ (here $a=2,\; b=3,\; c=-7$).
• $x - 4y = 0$ (here $a=1,\; b=-4,\; c=0$).
• $5x = 3$ can be written as $5x + 0 \cdot y - 3 = 0$ (linear in two variables, with $b = 0$).
• $y = 7$ can be written as $0 \cdot x + y - 7 = 0$ (linear in two variables, with $a = 0$).
• $\sqrt{2}\,x + y = 3$ — valid, since coefficients can be irrational.

Non-examples: $x^2 + y = 4$ (power of $x$ is 2, not linear); $xy = 5$ (product of variables — not linear); $\sqrt{x} + y = 1$ (variable under a root).

2. Solutions — ordered pairs, infinitely many

A solution of a linear equation $ax + by + c = 0$ is an ordered pair $(x_0, y_0)$ such that substituting $x = x_0$ and $y = y_0$ satisfies the equation: $ax_0 + by_0 + c = 0$.

Why infinitely many solutions? For any value of $x$ you choose, you can always find a corresponding $y$ (when $b \neq 0$): $y = \dfrac{-(ax + c)}{b}$. Since we can choose $x$ to be any real number and there are infinitely many real numbers, there are infinitely many solutions.

Notation matters: A solution is an ordered pair $(x, y)$ — order is important. $(2, 3)$ and $(3, 2)$ are different ordered pairs and will generally not both satisfy the same equation.

3. Graphing a linear equation — table of values, plot, join

Since every solution $(x, y)$ corresponds to a point in the coordinate plane and all solutions lie on one straight line, we graph the equation by a simple three-step process:

1. Make a table of values: choose at least 3 values of $x$ (convenient integers make arithmetic easier), compute the corresponding $y$ to get ordered pairs.
2. Plot the points on a coordinate plane with labelled axes.
3. Join the points with a straight line, and extend it in both directions with arrowheads to show it continues indefinitely.

Why 3 points? Two points determine a unique line. The third point acts as a check — if all three are collinear, your arithmetic is likely correct. If they are not collinear, re-check your table.

Choosing easy values: pick $x = 0$ to get the $y$-intercept, pick $y = 0$ to get the $x$-intercept. These two axis intercepts are the most natural plotting points.

Fundamental facts about the graph of $ax + by + c = 0$:

• Always a straight line — never a curve.
• Every point on the line is a solution; every solution is a point on the line.
• The line extends infinitely in both directions — there are infinitely many solutions.
• Draw arrowheads at both ends of the line segment to indicate this.

4. Graph passes through the origin when $c = 0$

If the constant term $c = 0$, the equation becomes $ax + by = 0$. Substituting $x = 0,\; y = 0$: $a(0) + b(0) = 0$. This is always true, so $(0, 0)$ is always a solution. Therefore, the graph passes through the origin.

Conversely, if $c \neq 0$, then $a(0) + b(0) + c = c \neq 0$, so the origin is not on the line.

Rule: Graph of $ax + by + c = 0$ passes through the origin $\Leftrightarrow$ $c = 0$.

Memory trick: "No constant — through the origin." If you see no constant term in the equation (or the constant is zero after rearranging), the line goes through $(0, 0)$.

5. Equations of lines parallel to the $y$-axis: $x = k$

The equation $x = k$ (where $k$ is a real constant) means that the $x$-coordinate is always equal to $k$, regardless of the value of $y$. In standard form: $1\cdot x + 0 \cdot y - k = 0$, so $a = 1,\; b = 0,\; c = -k$.

Solutions: $(k, 0),\; (k, 1),\; (k, -1),\; (k, 5),\ldots$ — all pairs where the first coordinate equals $k$.

Graph: All these points have the same $x$-coordinate, so they lie on a vertical line passing through the point $(k, 0)$ on the $x$-axis. This line is parallel to the $y$-axis.

• $k = 0$: the equation $x = 0$ is the $y$-axis itself.
• $k > 0$: vertical line to the right of the $y$-axis.
• $k < 0$: vertical line to the left of the $y$-axis.

All lines of the form $x = k$ are parallel to each other and parallel to the $y$-axis (and hence perpendicular to the $x$-axis).

6. Equations of lines parallel to the $x$-axis: $y = k$

The equation $y = k$ means the $y$-coordinate is always $k$, and $x$ can be anything. In standard form: $0 \cdot x + 1 \cdot y - k = 0$, so $a = 0,\; b = 1,\; c = -k$.

Solutions: $(0,\,k),\; (1,\,k),\; (-3,\,k),\ldots$ — all pairs where the second coordinate equals $k$.

Graph: All points have the same $y$-coordinate — a horizontal line at height $k$, parallel to the $x$-axis.

• $k = 0$: the equation $y = 0$ is the $x$-axis itself.
• $k > 0$: horizontal line above the $x$-axis.
• $k < 0$: horizontal line below the $x$-axis.

Summary table — special lines in the plane:

7. Representing real-life situations as linear equations

Many everyday relationships between two quantities can be expressed as linear equations in two variables. The process:

1. Identify the two unknown quantities and assign variable names ($x$ and $y$).
2. Translate the given condition into an algebraic equation using $x$ and $y$.
3. Simplify and write in standard form $ax + by + c = 0$.

Important note: when an equation models a real situation, not all mathematical solutions are physically meaningful. For example, negative values of $x$ or $y$ representing lengths or counts are rejected. We restrict solutions to those satisfying the real-life constraints.

8. Common mistakes to avoid

• Reversing the ordered pair: $(2, 3)$ and $(3, 2)$ are different. Always write $(x, y)$ — $x$ first, $y$ second.
• Forgetting arrowheads: the line extends infinitely — always draw arrowheads at both ends on the graph.
• Drawing a curve: linear equation always gives a straight line — never draw a curve.
• $x = k$ is parallel to the $x$-axis? No! $x = k$ is vertical, hence parallel to the $y$-axis.
• Saying "one solution" or "two solutions": a linear equation in two variables always has infinitely many solutions.
• Missing $c = 0$ check: only check whether the graph passes through the origin by checking whether the constant term is zero.
• Checking only two points: always use a third point to verify your table before drawing the line.
• Sign errors: when rearranging to standard form, change signs carefully when moving terms across the equals sign.

9. Quick revision checklist

• Standard form: $ax + by + c = 0$, with $a$ and $b$ not both zero.
• Every linear equation in two variables has infinitely many solutions.
• A solution is an ordered pair $(x, y)$ satisfying the equation — substitute both values to verify.
• Graph = straight line; build a table of at least 3 values, plot, then join with arrowheads.
• Use $y = 0$ to find the $x$-intercept; use $x = 0$ to find the $y$-intercept.
• $c = 0$ means the line passes through the origin $(0, 0)$.
• $x = k$ is a vertical line, parallel to the $y$-axis; $x = 0$ is the $y$-axis itself.
• $y = k$ is a horizontal line, parallel to the $x$-axis; $y = 0$ is the $x$-axis itself.
• Real-life situations: identify two unknowns, write one equation, graph it.