Grade 10 Linear Equations in Two Variables

Linear Equation: An equation in which the highest power of the variable(s) is 1.

Two Variables: The equation involves two different variables, usually 'x' and 'y'.

Standard Form: A linear equation in two variables can be written in the form ax + by + c = 0, where 'a', 'b', and 'c' are real numbers, and 'a' and 'b' are not both zero.

Example: 2x + 3y - 5 = 0 is a linear equation in two variables (a=2, b=3, c=-5).
x - 4y = 7 can be written as x - 4y - 7 = 0 (a=1, b=-4, c=-7).

Non-Examples:
x² + y = 5 (not linear, x has power 2)
xy + y = 2 (not linear, xy term)
3x = 6 (linear, but only one variable)

Solution: A pair of values for x and y that satisfies the equation (makes the equation true) is called a solution to the linear equation.

Ordered Pair: A solution is written as an ordered pair (x, y).

Example: Consider the equation x + y = 5.

• If x = 2 and y = 3, then 2 + 3 = 5. This is true. So, (2, 3) is a solution.
• If x = 1 and y = 4, then 1 + 4 = 5. This is true. So, (1, 4) is a solution.
• If x = 6 and y = 1, then 6 + 1 = 7, which is not equal to 5. So, (6, 1) is NOT a solution.

Infinite Solutions: A single linear equation in two variables has infinitely many solutions. Each solution corresponds to a point on the line represented by the equation on a graph.

Pair of Equations: When we have two linear equations with the same two variables, they form a system or a pair of linear equations.

Solving the Pair: Finding the values of x and y that satisfy BOTH equations simultaneously. This common solution represents the point where the graphs of the two lines intersect.

Methods: There are several methods to solve a pair of linear equations:

• Graphical Method
• Substitution Method
• Elimination Method
• Cramer's Rule (Determinant Method)

Example: Solve the pair:
Equation 1: x + y = 5
Equation 2: x - y = 1

Using Elimination Method: Add the two equations.
(x + y) + (x - y) = 5 + 1
2x = 6
x = 3

Substitute x=3 into Equation 1:
3 + y = 5
y = 5 - 3
y = 2

Result: The solution is (x, y) = (3, 2). This pair satisfies both equations (3+2=5 and 3-2=1).

Consistent System: A pair of linear equations is consistent if it has at least one solution. The graphs of the lines intersect at one point (unique solution) or coincide (infinitely many solutions).

Inconsistent System: A pair of linear equations is inconsistent if it has no solution. The graphs of the lines are parallel and never intersect.

Checking Consistency (using coefficients): For a system a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0:

• If a1/a2 $\neq$ b1/b2: Consistent (Unique Solution, Intersecting Lines)
• If a1/a2 = b1/b2 = c1/c2: Consistent (Infinitely Many Solutions, Coinciding Lines)
• If a1/a2 = b1/b2 $\neq$ c1/c2: Inconsistent (No Solution, Parallel Lines)