Chapter 4 — Linear Equations In Two Variables

Definition: A linear equation in two variables is an equation of the form ax + by + c = 0, where a and b are not both zero, and x and y are variables. Often written as ax + by = c.

What is a solution? A solution is an ordered pair (x, y) that satisfies the equation. Unlike linear equations in one variable, a linear equation in two variables has infinitely many solutions (they form a set of ordered pairs).

Geometric meaning: The complete set of solutions of a linear equation in two variables corresponds to all the points lying on a straight line in the Cartesian plane. Thus every linear equation in two variables represents a straight line.

How to find solutions:

• Give any value to one variable (say x) and solve for the other variable (y) to get a solution pair.
• To draw the line, it is enough to find two distinct solutions (two points) and join them.

Slope and intercepts: If the equation is written as ax + by = c (with b ≠ 0), we can write y = (-a/b)x + c/b. This is the slope-intercept form y = mx + k where m = -a/b is the slope and k = c/b is the y-intercept. The x-intercept is obtained by putting y = 0, giving x = c/a (if a ≠ 0).

Special cases:

• If b = 0 and a ≠ 0, the equation reduces to x = constant, which is a vertical line.
• If a = 0 and b ≠ 0, the equation reduces to y = constant, which is a horizontal line.

Key ideas to remember: A single linear equation in two variables has infinitely many solutions; graph is a straight line; two points determine the line; slope tells steepness and sign tells direction.

What is a Cartesian plane? The Cartesian plane (or coordinate plane) is formed by two number lines that intersect at right angles. The horizontal number line is called the x-axis and the vertical number line is called the y-axis. Their intersection point is the origin, labeled O and having coordinates (0, 0). The plane is named after René Descartes.

Coordinates and ordered pair Every point P in the plane is specified by an ordered pair (x, y). The first number x is the abscissa (horizontal distance from the origin) and the second number y is the ordinate (vertical distance from the origin). Follow the order: x first, then y.

Quadrants and sign conventions The axes divide the plane into four quadrants numbered I to IV counterclockwise from the top-right. Sign of coordinates in each quadrant: quadrant I: ( + , + ), quadrant II: ( - , + ), quadrant III: ( - , - ), quadrant IV: ( + , - ). Points on the x-axis have y = 0; points on the y-axis have x = 0.

How to plot a point (step-by-step) 1) Start at the origin. 2) Move horizontally by x units: right if x is positive, left if x is negative. 3) From there move vertically by y units: up if y is positive, down if y is negative. 4) Mark the point and label it with its ordered pair.

Using the Cartesian plane for linear equations A linear equation in two variables (for example y = mx + c) represents a straight line on the Cartesian plane. To draw it: choose several x-values, compute corresponding y-values to get points, plot those points and join them to form a straight line. Vertical lines have equations x = a and horizontal lines have equations y = b.

Practical tips Always choose a scale and mark equal unit lengths on both axes. Label axes and the origin. Use at least two distinct points to determine a straight line, but plotting three helps verify accuracy.

Definition: A linear equation in two variables is an equation of the form ax + by + c = 0 (where a and b are not both zero). A solution is an ordered pair (x, y) which, when substituted into the equation, makes it true.

Key ideas:

• Every solution is an ordered pair (x, y). Example: for 2x + 3y = 12, (3, 2) is a solution because 2(3)+3(2)=12.
• A linear equation in two variables has infinitely many solutions. These solutions form a straight line in the xy-plane.
• To find solutions: pick a value for one variable and solve for the other. Usually we make a small table of values and plot the points.
• Special cases:
  • If a = b = 0 and c = 0, every ordered pair (x,y) is a solution (infinitely many).
  • If a = b = 0 and c ≠ 0, there is no solution.

Method (step-by-step):

1. Write equation in the form ax + by + c = 0 or y = mx + c.
2. Choose convenient values for x (or y) such as -2, -1, 0, 1, 2.
3. Compute the corresponding y (or x) to make ordered pairs.
4. Plot at least three points and draw the straight line through them — every point on this line is a solution.

Relation to slope-intercept form: If ax + by + c = 0 and b ≠ 0, then y = (-a/b)x + (-c/b). Here slope m = -a/b and y-intercept = -c/b. If b = 0 (equation ax + c = 0) the graph is a vertical line x = -c/a; if a = 0 (by + c = 0) the graph is a horizontal line y = -c/b.

What it is: A linear equation in two variables x and y is an equation of the form ax + by + c = 0 (where a and b are not both zero). The graph of such an equation is the set of all ordered pairs (x,y) that satisfy the equation. All these points lie on a straight line in the Cartesian plane, so the graph is a straight line.

Key ideas:

• Each solution (x,y) is a point on the line. The line contains infinitely many solutions.
• To draw the line it is enough to find two distinct solutions (two points) and join them by a straight edge. (Use more points to verify accuracy.)
• Special lines: if b = 0 the line is vertical (x = constant); if a = 0 the line is horizontal (y = constant).

Procedure to graph a linear equation:

1. Write the equation in a convenient form, e.g. y = mx + c (if possible).
2. Make a table of values: choose two or three x-values, compute corresponding y-values.
3. Plot the computed points (x,y) on the Cartesian plane.
4. Join the points with a ruler to get the straight line; extend it and draw arrows at ends to show it continues.

Intercepts and slope:

• x-intercept: point where the line meets x-axis (set y = 0 and solve for x).
• y-intercept: point where the line meets y-axis (set x = 0 and solve for y). The y-intercept is (0, c) if equation written y = mx + c.
• Slope (m) measures steepness: for two points (x1,y1) and (x2,y2), m = (y2 - y1)/(x2 - x1). In ax + by + c = 0 form, slope = -a/b (provided b ≠ 0).

Properties to note:

• If the constant term is zero (ax + by = 0), the line passes through the origin.
• Parallel lines have equal slopes. Perpendicular lines have slopes whose product is -1 (if both slopes are defined).

Quick example of graphing (method): To graph y = 2x + 1, pick x = 0 → y = 1 gives (0,1); pick x = 1 → y = 3 gives (1,3). Plot (0,1) and (1,3) and join to draw the line.

Definition. For a line in the coordinate plane, the x-intercept is the x-coordinate of the point where the line meets the x-axis (y = 0). The y-intercept is the y-coordinate of the point where the line meets the y-axis (x = 0).

Intercept Form. If a line cuts the x-axis at (a, 0) and the y-axis at (0, b), and both a and b are finite and nonzero, the equation of the line can be written in intercept form:

x/a + y/b = 1

This form shows immediately the intercepts: x-intercept = a and y-intercept = b.

Derivation from two intercepts. The line joining (a,0) and (0,b) satisfies every point (x,y) that is a convex combination of the intercept points: x = a(1 - t), y = b t for some t. Eliminating t gives x/a + y/b = 1.

Conversion between general form and intercept form. For a general straight line ax + by + c = 0 (with a ≠ 0 and b ≠ 0), the intercepts are:

• x-intercept = -c/a (put y = 0)
• y-intercept = -c/b (put x = 0)

If these intercepts are A = -c/a and B = -c/b (both nonzero), the equation can be written as x/A + y/B = 1.

Slope from intercept form. From x/a + y/b = 1, rearrange to y = b - (b/a)x, so the slope m = -b/a (note: here a and b are the intercepts in the intercept form).

Special cases & limitations.

• If the line is vertical (x = k), it does not meet the y-axis at a finite point so intercept form x/a + y/b = 1 is not applicable.
• If the line is horizontal (y = k), intercept form with finite a is not applicable in the usual way because one intercept would be infinite.
• If c = 0 in ax + by + c = 0 then the line passes through the origin; both intercepts are 0 and the intercept form is not meaningful.

How to use intercept form to graph a line. 1) Read the intercepts a and b from x/a + y/b = 1. 2) Plot (a, 0) and (0, b) on the axes. 3) Join the two points with a straight line. That line is the graph of the equation.

What is slope?
The slope of a straight line is a number that measures its steepness and direction. It is often denoted by m. Given two distinct points on the line (x₁, y₁) and (x₂, y₂), the slope is

m = (y₂ − y₁) / (x₂ − x₁)

Interpretation: rise/run — how much y changes when x changes by 1. Types:

• Positive slope (m > 0): line rises left to right.
• Negative slope (m < 0): line falls left to right.
• Zero slope (m = 0): horizontal line (no rise).
• Undefined slope: vertical line (x = constant).

Slope-intercept form
The slope-intercept form of the equation of a line is

y = mx + c

Here m is the slope and c is the y-intercept (the point where the line crosses the y-axis). From this form you can immediately read the slope and the intercept:

• Slope = m
• y-intercept = (0, c)

Converting other forms to slope-intercept form
If a line is given in standard form ax + by + d = 0, solve for y:

y = (−a/b) x + (−d/b)

So slope m = −a/b and y-intercept c = −d/b (provided b ≠ 0).

Relations

• Two lines are parallel if they have equal slopes (m₁ = m₂).
• Two non-vertical lines are perpendicular if m₁·m₂ = −1 (each slope is the negative reciprocal of the other).

How to graph a line in slope-intercept form y = mx + c

1. Plot the y-intercept (0, c) on the y-axis.
2. From that point use the slope m = rise/run: move right by 1 (or run) and up/down by rise; if run is not 1, scale accordingly.
3. Plot a second point and draw the straight line through the points.

Example visual cues: mark the y-intercept, draw a right triangle under the line showing rise and run, and label the slope as rise/run.

What is a linear equation in two variables? A linear equation in two variables x and y is of the form ax + by + c = 0 (or ax + by = c) where a and b are not both zero. Its solution set is all ordered pairs (x, y) that lie on the straight line represented by that equation.

Special cases (geometric meaning)

• One unique solution: Two lines intersect at exactly one point. Algebraically, for equations a1x + b1y = c1 and a2x + b2y = c2, if a1/a2 ≠ b1/b2, the lines are not parallel and have exactly one common point (unique solution).
• No solution (inconsistent): Two lines are parallel and distinct, so they never meet. This happens when a1/a2 = b1/b2 but a1/a2 ≠ c1/c2.
• Infinitely many solutions (dependent): Two equations represent the same line (coincident). This occurs when a1/a2 = b1/b2 = c1/c2. Every point on the line is a solution.

Alternative (determinant) test: For the pair a1x + b1y = c1 and a2x + b2y = c2 define D = a1 b2 - a2 b1.

• If D ≠ 0 → unique solution (lines intersect).
• If D = 0 and Dx = 0 and Dy = 0 → infinitely many solutions (coincident lines).
• If D = 0 and (Dx ≠ 0 or Dy ≠ 0) → no solution (parallel lines).

Slope and intercepts (useful for graphs)

• Write ax + by = c as y = (-a/b)x + (c/b). The slope m = -a/b and y-intercept = c/b (if b ≠ 0).
• For form ax + by + c = 0, slope m = -a/b and y-intercept = -c/b; x-intercept = -c/a (when a ≠ 0, b ≠ 0 respectively).

Practical note: If c = 0 (equation ax + by = 0) the line passes through the origin. If b = 0 the line is vertical (x = constant); if a = 0 the line is horizontal (y = constant).

Applications and word problems using linear equations in two variables teach how to model real-life situations with equations of the form ax + by = c, interpret their solutions, and solve for the unknown quantities. Each problem requires translating a verbal description into one or more linear equations, choosing variables, solving the system (graphically or algebraically), and interpreting the results in context.

Key ideas:

• Modeling: Assign variables to the unknown quantities (for example x = number of adult tickets, y = number of child tickets) and convert statements into linear equations.
• Single linear equation in two variables represents a line. Its set of solutions is all ordered pairs (x,y) on that line (useful when the question asks for relation between quantities).
• Two linear equations in two variables form a system. The intersection of the two lines gives the solution: one intersection = unique solution; parallel lines = no solution; coincident lines = infinitely many solutions.
• Methods of solution: substitution, elimination (addition/subtraction), graphing, or determinant/Cramer’s rule for 2×2 systems.
• Always check that the algebraic solution makes sense in the real-life context (e.g., counts must be integers and non‑negative).

Steps to solve a word problem:

1. Read carefully and identify what is asked.
2. Choose variables and write them clearly (x = ... , y = ...).
3. Translate the sentences into one or two linear equations.
4. Solve the equations by substitution or elimination (or graphically if asked).
5. Interpret and check the solution in the original context.

Special cases to watch for:

• If the two equations are proportional (a1/a2 = b1/b2 = c1/c2) the lines coincide → infinitely many solutions.
• If a1/a2 = b1/b2 but c1/c2 differs, the lines are parallel → no solution.

Verification: A pair (x, y) is a solution of a linear equation in two variables if, on substituting x and y in the equation, the equality holds. Example: to verify whether (2, 3) is a solution of 2x + 3y = 13, substitute: 2(2) + 3(3) = 4 + 9 = 13, so (2, 3) is a solution.

Discussion: Key properties of a linear equation in two variables (general form ax + by + c = 0 or ax + by = k):

• Solutions are ordered pairs (x, y). There are infinitely many solutions—each solution corresponds to a point on a straight line (the graph of the equation).
• Two points are enough to draw the line: pick any two distinct solutions, plot them and join to get the graph representing all solutions.
• Special cases: if a = 0 (equation becomes by + c = 0) it is a horizontal line; if b = 0 it is a vertical line.
• For a system of two linear equations in two variables, there are three possibilities: a unique solution (lines intersect at one point), no solution (parallel lines), or infinitely many solutions (coincident lines).

Summary / Practical points:

• To verify: substitute candidate (x, y) into the equation and check equality.
• To graph: rewrite into y = mx + c form (if possible), find intercepts or two convenient points, then draw the line.
• Intercepts: x-intercept is obtained by putting y = 0; y-intercept by putting x = 0.
• For two lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, compare ratios a1/a2, b1/b2, c1/c2 to decide intersection/parallel/coincident.