A linear equation in two variables is of the form ax + by + c = 0, where a, b, c are real numbers and a and b are not both zero. Its graph is always a straight line.
Solutions: An ordered pair (x, y) satisfying the equation is called a solution. A linear equation in two variables has infinitely many solutions, each corresponding to a point on its graph.
Graph of a linear equation: To draw the graph, find at least two solutions, plot them, and draw a straight line through them.
- Special cases:
- If b = 0 (equation like x = k), the graph is a vertical line parallel to the y-axis.
- If a = 0 (equation like y = k), the graph is a horizontal line parallel to the x-axis.
- y = mx passes through the origin with slope m.
Write three solutions of 2x + y = 7.
Set x = 0: y = 7 → (0, 7).
Set x = 1: y = 5 → (1, 5).
Set x = 3: y = 1 → (3, 1).
Does (2, 3) lie on the line x + 2y = 8?
LHS = 2 + 2(3) = 2 + 6 = 8 = RHS. Yes, the point satisfies the equation.
Find the graph of x = 3.
This is a vertical line passing through all points (3, y) for any y. It is parallel to the y-axis.
Express y = 5 in the form ax + by + c = 0.
0 · x + 1 · y - 5 = 0, so a = 0, b = 1, c = -5.
If (k, 2) is a solution of 3x - 4y = 10, find k.
3k - 4(2) = 10 → 3k - 8 = 10 → 3k = 18 → k = 6.
Graph 2x + 3y = 12 and find where it meets the axes.
x-intercept: set y = 0, 2x = 12, x = 6. Point: (6, 0).
y-intercept: set x = 0, 3y = 12, y = 4. Point: (0, 4).
Plot (6, 0) and (0, 4) and join them.
A sum of two numbers is 20. Write as a linear equation and find two solutions.
Let the numbers be x and y: x + y = 20.
Solutions: (8, 12) and (15, 5).
Common mistakes
Students sometimes think a linear equation has only one solution. Remember that it has infinitely many. Also, be careful with equations like x = 5: this is valid in two variables (representing the vertical line x = 5) even though y does not appear.
Summary
Linear equations in two variables have infinitely many solutions forming a straight line on the Cartesian plane. Any point on the line is a solution; any point off the line is not. Intercepts are a convenient way to draw the graph.