0What is a Linear Equation in Two Variables?
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A linear equation in two variables is an equation that involves two unknowns (usually x and y) and represents a straight line on a graph.
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Each solution of the equation is a pair of numbers that satisfies the equation.
2. Solutions of a Linear Equation in Two Variables
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A solution is a pair of values for the two variables that makes the equation true.
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There are infinite solutions for a linear equation in two variables because many pairs of numbers can satisfy it.
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Solutions can be represented as points on a graph.
3. Graphical Representation
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Each solution of the equation corresponds to a point on the coordinate plane.
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All points representing solutions form a straight line.
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So, a linear equation in two variables always forms a straight line.
4. Types of Solutions
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Unique Solution (One Point)
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Two lines intersect at one point → one solution (x, y).
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Infinite Solutions (Coinciding Lines)
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Two lines overlap completely → all points on the line are solutions.
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No Solution (Parallel Lines)
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Two lines are parallel and never meet → no common solution.
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5. Important Concepts
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Intercepts: Points where the line crosses X-axis and Y-axis.
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Slope (conceptual): Steepness of the line; slope shows how fast one variable changes with respect to the other.
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Plotting Method: Pick two points from the solutions and draw a straight line through them.
6. Key Notes for Exams
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Linear equation in two variables → represents a straight line.
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Each solution = ordered pair (x, y).
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Infinite solutions → points on a line.
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Graph is the visual representation of solutions.
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Intercepts help in drawing the graph quickly.
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Solutions of two equations can be unique, infinite, or none.
