Linear Equations in Two Variables: Graphs and Solutions

A linear equation in two variables has the form ax + by + c = 0, where a and b are not both zero. Unlike a one-variable equation, it has infinitely many solutions, and every solution is an ordered pair (x, y). Plotted on a graph, all these pairs lie on a single straight line.

This matters because most real relationships involve two quantities changing together: cost versus items bought, distance versus time, temperature versus altitude. The line is the picture of the relationship, and reading the line answers practical questions instantly.

Key Concepts

• Ordered pair: a solution written as (x, y), where the order matters: x first, then y.
• Standard form: ax + by + c = 0, useful for finding intercepts.
• Slope-intercept form: y = mx + c, where m is the slope (steepness) and c is the y-intercept. For ax + by + c = 0, the slope is -a/b.
• Intercepts: where the line crosses the axes. Set x = 0 for the y-intercept and y = 0 for the x-intercept.

How It Works

1. Find the y-intercept by setting x = 0 and solving for y.
2. Find the x-intercept by setting y = 0 and solving for x.
3. Plot these two points and draw a straight line through them; every point on the line is a solution.
4. Alternatively, rearrange into y = mx + c to read off the slope and intercept directly.

Worked Example

Graph 2x + 3y = 12 and describe its solutions.

1. Set x = 0: 3y = 12, so y = 4. The y-intercept is (0, 4).
2. Set y = 0: 2x = 12, so x = 6. The x-intercept is (6, 0).
3. Join (0, 4) and (6, 0) with a straight line.
4. Rearranging: y = 4 - (2/3)x, so the slope is -2/3 and every solution satisfies this relation.

Test another point on the line, say x = 3: y = 4 - 2 = 2, and 2(3) + 3(2) = 6 + 6 = 12. It satisfies the equation, confirming the line is correct.

Common Misconceptions

• A two-variable equation has a single solution. Wrong. It has infinitely many solutions, one for each point on its line.
• The slope of ax + by + c = 0 is a/b. Wrong. Rearranging gives y = -(a/b)x - c/b, so the slope is -a/b. For 2x + 3y = 12 the slope is -2/3, not 2/3.
• One plotted point is enough to draw the line. Wrong. A single point does not fix a direction; you need at least two points, or one point plus the slope.

Real-World Applications

• In business, total cost = fixed cost + (variable cost times quantity) is a two-variable linear equation used for break-even analysis.
• In physics, distance-time graphs for uniform motion are straight lines whose slope is the speed.
• In economics, simple supply and demand relationships are modelled as lines, and their intersection gives the equilibrium price.

To recap: a linear equation in two variables represents a straight line of infinitely many solutions. Find two intercepts, plot them, and read the slope from y = mx + c form, remembering the negative sign in -a/b.