Class 9 Mathematic Linear Equations in Two Variables All Formula & Concept Notes | Formula Sheet

1. Introduction to Linear Equations in Two Variables

• Definition: A linear equation in two variables is an equation of the form:

  $$ax + by + c = 0$$

  where $x$ and $y$ are variables, and $a$, $b$, and $c$ are constants. The highest power of the variables is 1, making the equation linear.

• General Form: The general form of a linear equation in two variables is:

  $$ax + by = c$$

  Here, $a$ and $b$ are not both zero.

2. Solution of a Linear Equation

• Solution: A solution of a linear equation is a pair of values, one for $x$ and one for $y$, which makes the equation true.

• Example: Consider the equation $2x + 3y = 6$.

  • A solution could be $x = 0$ and $y = 2$ because substituting these values satisfies the equation: $2(0) + 3(2) = 6$.

3. Graph of a Linear Equation

• Plotting: To graph a linear equation, we find two or more solutions of the equation and plot these points on the coordinate plane. The line passing through these points is the graph of the equation.

• Steps to Plot:

  1. Convert the equation to $y = mx + c$ form (if possible), where $m$ is the slope and $c$ is the y-intercept.
  2. Find at least two solutions (points) by substituting different values of $x$ and solving for $y$.
  3. Plot these points on the coordinate plane.
  4. Draw a line through these points.

• Example: For the equation $x + y = 4$,

  • Solutions: $(0, 4)$ and $(4, 0)$.
  • Plot these points and draw a line through them.

4. Intercepts

• X-intercept: The point where the graph of the equation crosses the x-axis ($y = 0$).
  • Find by setting $y = 0$ and solving for $x$.
• Y-intercept: The point where the graph of the equation crosses the y-axis ($x = 0$).
  • Find by setting $x = 0$ and solving for $y$.

5. Slope of a Line

• Definition: The slope (m) of a line represents the steepness and direction of the line. It is the ratio of the change in $y$ to the change in $x$ between two points on the line.

  $$m = \frac{y_2 - y_1}{x_2 - x_1}$$

  where $(x_1, y_1)$ and $(x_2, y_2)$ are two points on the line.

• Positive Slope: If $m > 0$, the line rises as it moves from left to right.

• Negative Slope: If $m < 0$, the line falls as it moves from left to right.

6. Types of Linear Equations

• Horizontal Line: The equation is of the form $y = c$, where $c$ is a constant. The slope $m = 0$.
• Vertical Line: The equation is of the form $x = c$, where $c$ is a constant. The slope is undefined.

7. Conditions for Consistency

• Unique Solution: If two lines intersect at a point, the system of equations has a unique solution.
• No Solution: If two lines are parallel, they do not intersect, and the system of equations has no solution.
• Infinite Solutions: If two lines coincide, the system of equations has infinitely many solutions.

8. Applications

• Word Problems: Linear equations are often used to model real-life situations. Common applications include problems related to motion, mixtures, and financial calculations.

• Example: A problem could involve finding the cost of items given the total cost and quantity.

9. Important Formulae

1. Standard Form of a Linear Equation: $ax + by = c$
2. Slope-Intercept Form: $y = mx + c$, where $m$ is the slope and $c$ is the y-intercept.
3. Point-Slope Form: $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line.
4. Two-Point Form: $y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)$, used when two points are known.
5. Intercept Form: $\frac{x}{a} + \frac{y}{b} = 1$, where $a$ and $b$ are the x- and y-intercepts, respectively.

10. Practice Problems

1. Find the solution to the equation $3x - 4y = 12$.
2. Determine the x- and y-intercepts for the equation $5x + 7y = 35$.
3. Plot the graph of the equation $x - 2y = 3$.
4. Solve the system of equations: $$\begin{align*} 2x + 3y &= 12 \\ 4x - y &= 8 \end{align*}$$

11. Summary

Understanding linear equations in two variables involves knowing how to interpret, solve, and graph equations. Mastery of these concepts provides a strong foundation for more advanced algebraic concepts.