LINEAR EQUATIONS IN TWO VARIABLES

1.1 Linear Equation in One Variable (Revision)

• An equation of the form \(ax + b = 0\), where \(a, b\) are real numbers and \(a \neq 0\), is called a linear equation in one variable.
• It has exactly one unique solution.
• Example: \(2x + 5 = 0 \Rightarrow 2x = -5 \Rightarrow x = -5/2\).

1.2 Linear Equation in Two Variables

• An equation of the form \(ax + by + c = 0\), where \(a, b, c\) are real numbers, and \(a \neq 0\) and \(b \neq 0\), is called a linear equation in two variables.
• Here, \(x\) and \(y\) are the two variables.
• The highest power of each variable is 1.
• Examples:
• \(2x + 3y = 5\)
• \(4p - 7q = 12\)
• \(x - \sqrt{2}y = 4\)

1.3 Standard Form of a Linear Equation in Two Variables

• The standard form is \(ax + by + c = 0\).
• Any linear equation in two variables can be written in this form.
• Identifying a, b, c:
• \(a\) is the coefficient of x.
• \(b\) is the coefficient of y.
• \(c\) is the constant term.
• Example 1: Express \(2x = 5y - 7\) in the form \(ax + by + c = 0\) and find \(a, b, c\).
• Step 1: Rearrange the terms: \(2x - 5y + 7 = 0\)
• Step 2: Compare with \(ax + by + c = 0\).
• Result: \(a = 2, b = -5, c = 7\).
• Example 2: Express \(y = 2x\) in the form \(ax + by + c = 0\) and find \(a, b, c\).
• Step 1: Rearrange: \(2x - y + 0 = 0\)
• Step 2: Compare.
• Result: \(a = 2, b = -1, c = 0\).
• Example 3: Express \(5 = 2x\) as a linear equation in two variables.
• Step 1: Add the \(y\) term with coefficient 0: \(2x + 0y - 5 = 0\)
• Result: \(a = 2, b = 0, c = -5\).
• _Note: Here, \(b=0\) is allowed, but for a true 'two variable' equation, usually \(a \neq 0\) and \(b \neq 0\) are specified. If only one variable is present, it's often treated as a special case of two variables._

2.1 What is a Solution?

• A solution to a linear equation in two variables \(ax + by + c = 0\) is a pair of values \((x, y)\) that makes the equation a true statement.
• This pair \((x, y)\) is also called an ordered pair.

2.2 How to Find Solutions

• Since there are two variables, we can choose a value for one variable and then solve for the other.
• Steps:

1. Rearrange the equation to express one variable in terms of the other (e.g., \(y = mx + k\) or \(x = ny + k\)).
2. Choose any real value for the independent variable (e.g., \(x\)).
3. Substitute this value into the rearranged equation to find the corresponding value of the dependent variable (e.g., \(y\)).
4. The pair \((x, y)\) is a solution.

• Example: Find four solutions for the equation \(2x + y = 6\).
• Step 1: Express \(y\) in terms of \(x\): \(y = 6 - 2x\).
• Step 2: Choose values for \(x\):
• If \(x = 0\), then \(y = 6 - 2(0) = 6\). Solution: \((0, 6)\).
• If \(x = 1\), then \(y = 6 - 2(1) = 4\). Solution: \((1, 4)\).
• If \(x = 2\), then \(y = 6 - 2(2) = 2\). Solution: \((2, 2)\).
• If \(x = 3\), then \(y = 6 - 2(3) = 0\). Solution: \((3, 0)\).

2.3 Infinite Solutions

• A linear equation in two variables has infinitely many solutions.
• This is because we can choose any real number for one variable, and there will always be a corresponding real number for the other variable that satisfies the equation.
• Each solution represents a point on the line that the equation represents.

2.4 Verifying a Solution

• To check if a given pair \((x_0, y_0)\) is a solution, substitute these values into the equation.
• If LHS = RHS, then it is a solution.
• Example: Check if \((2, 1)\) is a solution to \(3x - 2y = 4\).
• LHS = \(3(2) - 2(1) = 6 - 2 = 4\).
• RHS = \(4\).
• Since LHS = RHS, \((2, 1)\) is a solution.
• Example: Check if \((1, 3)\) is a solution to \(x + y = 5\).
• LHS = \(1 + 3 = 4\).
• RHS = \(5\).
• Since LHS \(\neq\) RHS, \((1, 3)\) is NOT a solution.

3.1 What does the Graph Represent?

• Every linear equation in two variables, \(ax + by + c = 0\), represents a straight line on the Cartesian plane.
• Every point \((x, y)\) on this line is a solution to the equation.
• Conversely, every solution \((x, y)\) of the equation is a point on the line.

3.2 Steps to Draw the Graph

1. Find at least two solutions \((x_1, y_1)\) and \((x_2, y_2)\) for the given equation.

• _It's advisable to find a third solution to cross-check for accuracy. If all three points are collinear, your calculations are likely correct._

1. Plot these points on a Cartesian plane.
2. Draw a straight line passing through these points. Extend the line in both directions.

• Example: Draw the graph of \(x + y = 4\).
• Step 1: Find solutions.
• If \(x = 0\), \(y = 4 - 0 = 4\). Point: \((0, 4)\).
• If \(y = 0\), \(x = 4 - 0 = 4\). Point: \((4, 0)\).
• If \(x = 1\), \(y = 4 - 1 = 3\). Point: \((1, 3)\).
• Step 2: Plot points. Plot \((0, 4)\), \((4, 0)\), and \((1, 3)\) on the graph paper.
• Step 3: Draw line. Draw a straight line passing through these points.

3.3 Special Cases

3.3.1 Equation of the form \(y = kx\)

• Equations of the form \(y = kx\) (or \(y = mx\)) always pass through the origin \((0, 0)\).
• Example: Graph \(y = 2x\).
• If \(x = 0, y = 0\). Point: \((0, 0)\).
• If \(x = 1, y = 2\). Point: \((1, 2)\).
• If \(x = -1, y = -2\). Point: \((-1, -2)\).

3.3.2 Equation of the form \(ax + by = 0\)

• These equations also pass through the origin \((0, 0)\).
• Example: Graph \(2x + 3y = 0\).
• If \(x = 0, y = 0\). Point: \((0, 0)\).
• If \(x = 3, 2(3) + 3y = 0 \Rightarrow 6 + 3y = 0 \Rightarrow 3y = -6 \Rightarrow y = -2\). Point: \((3, -2)\).
• If \(x = -3, 2(-3) + 3y = 0 \Rightarrow -6 + 3y = 0 \Rightarrow 3y = 6 \Rightarrow y = 2\). Point: \((-3, 2)\).

3.4 Finding a Point on the Graph

• If a point \((x_0, y_0)\) lies on the graph of an equation, it must satisfy the equation.
• Example: Find the value of \(k\) if the point \((2, 3)\) lies on the graph of \(3x + ky = 12\).
• Substitute \(x = 2\) and \(y = 3\) into the equation:

\(3(2) + k(3) = 12\) \(6 + 3k = 12\) \(3k = 12 - 6\) \(3k = 6\) \(k = 2\).

3.5 Word Problems and Equations

• Translating word problems into linear equations is a key skill.
• Steps:

1. Identify the unknown quantities and assign variables (e.g., \(x\) and \(y\)).
2. Formulate the relationship between these quantities as an equation.

• Example: The cost of a notebook is twice the cost of a pen. Write a linear equation in two variables to represent this statement.
• Let the cost of a notebook be \(₹x\).
• Let the cost of a pen be \(₹y\).
• According to the problem, \(x = 2y\).
• In standard form: \(x - 2y = 0\).

4.1 Equation of the x-axis

• Every point on the x-axis has its y-coordinate as 0.
• Therefore, the equation of the x-axis is \(y = 0\).

4.2 Equation of the y-axis

• Every point on the y-axis has its x-coordinate as 0.
• Therefore, the equation of the y-axis is \(x = 0\).

4.3 Lines Parallel to the x-axis

• A line parallel to the x-axis has a constant y-coordinate.
• The equation of a line parallel to the x-axis at a distance of \(k\) units from it is \(y = k\) or \(y = -k\).
• If \(k > 0\), the line is above the x-axis.
• If \(k < 0\), the line is below the x-axis.
• Example: Graph \(y = 3\).
• This means for any value of \(x\), \(y\) is always 3.
• Points: \((0, 3), (1, 3), (-2, 3)\).
• This will be a horizontal line passing through \((0, 3)\).
• Example: Graph \(y = -2\).
• This will be a horizontal line passing through \((0, -2)\).

4.4 Lines Parallel to the y-axis

• A line parallel to the y-axis has a constant x-coordinate.
• The equation of a line parallel to the y-axis at a distance of \(k\) units from it is \(x = k\) or \(x = -k\).
• If \(k > 0\), the line is to the right of the y-axis.
• If \(k < 0\), the line is to the left of the y-axis.
• Example: Graph \(x = 4\).
• This means for any value of \(y\), \(x\) is always 4.
• Points: \((4, 0), (4, 1), (4, -3)\).
• This will be a vertical line passing through \((4, 0)\).
• Example: Graph \(x = -1\).
• This will be a vertical line passing through \((-1, 0)\).

4.5 Representing in One Variable vs. Two Variables

• An equation like \(y = 3\) can be seen:
• As an equation in one variable: It's a point on the number line.
• As an equation in two variables: \(0x + 1y = 3\). It's a line parallel to the x-axis on the Cartesian plane.
• Similarly, \(x = -2\):
• As an equation in one variable: It's a point on the number line.
• As an equation in two variables: \(1x + 0y = -2\). It's a line parallel to the y-axis on the Cartesian plane.