LINEAR EQUATIONS IN TWO VARIABLES

Notes on Linear Equations in Two Variables

1. Definition of Linear Equation in Two Variables

A linear equation in two variables can be expressed in the form:

ax + by + c = 0

Where:

• a, b, c are real numbers, and
• a and b cannot both be equal to zero.

This equation describes a line in the Cartesian coordinate system.

2. Examples of Linear Equations

Some example formats of linear equations include:

• 2x + 3y = 4.37
• x - 4 = 3y
• 4 = 5x - 3y
• 2x = y
  Each of these equations can be rearranged into the general form ax + by + c = 0.

3. Finding the Equation of a Scenario

When translating real-world situations into linear equations, as in the cricket match example where two batsmen scored a total of 176 runs.
Letting:

• x be the runs scored by one batsman

• y be the runs scored by the other,
  We can express this information as:

  x + y = 176

This represents a linear equation in two variables with an infinite number of solutions.

4. Solutions of Linear Equations in Two Variables

A linear equation in two variables can have infinitely many solutions.
Each solution is a pair of values (x, y) that satisfies the equation. For example, for the equation 2x + 3y = 12, the pairs:

• (3, 2)
• (0, 4)
• (6, 0)
  All satisfy the equation.
  To find solutions, you can substitute values for either variable and solve for the other.

5. Graphing Linear Equations

The graphical representation of a linear equation in two variables is a straight line on the Cartesian plane.
Each point on the line corresponds to a solution of the equation.
For instance, the points determining the line for x + 2y = 6 can be derived from substituting values of x or y to find corresponding pairs (x, y).

6. Methods to Determine Solutions

A common and effective method is to set x = 0 to find the y-intercept and y = 0 to find the x-intercept:

• Setting x = 0 in 2x + 5y = 0 yields:
  5y = 0 → y = 0 (y-intercept)
• Setting y = 0 yields:
  2x = 0 → x = 0 (x-intercept) The two intercepts can then be used to draw the graph of the line.

7. Key Properties and Verification

• Any reorder or modification of terms, such as multiplying or adding the same value across both sides of the equation, will still lead to equivalent equations.
• You can check if a pair (x, y) is a solution by substituting back into the original equation. If it satisfies the equation, it is a solution.

8. Exercises and Practice

Problems and scenarios provided at the end of the chapter enable practice of identifying and creating equations, solving them, and further grasping the concept of linear equations in two variables.

1. Linear equations in two variables are of the form ax + by + c = 0.
2. Such equations can represent infinite solutions on a graph.
3. Graph of a linear equation is a straight line in the Cartesian plane.
4. Each solution can be derived by substituting one variable and solving for the other.
5. Equations can be rearranged for various forms but represent the same relationship.
6. A solution is a unique point (x, y) that satisfies the equation.
7. Intercepts can be found by setting one variable to zero to determine points on the graph.
8. Use scenarios (word problems) to form linear equations for practical understanding.