Linear Equations in Two Variables Class 9 ||Maths|| Chapter 4 Notes

Linear Equation in Two Variables ||Maths|| Chapter 4 Notes

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4.1 Introduction

• A linear equation in two variables involves two variables and forms a straight line when plotted on a coordinate plane.
• The general form of the equation is:$$ax+by+c=0$$Where:
  • $a$, $b$, and $c$ are constants (real numbers),
  • $x$ and $y$ are variables.

Example: $2x+3y=6$ is a linear equation in two variables.

4.2 Solutions of a Linear Equation

• A solution of a linear equation in two variables is a pair of values $(x,y)$ that satisfy the equation when substituted into it.
• There can be infinitely many solutions to a linear equation in two variables, and every solution represents a point on the line represented by the equation.

Example:

• For the equation $x+y=5$:
  • $(2,3)$ is a solution because $2+3=5$.
  • $(4,1)$ is another solution because $4+1=5$.

4.3 Graph of a Linear Equation in Two Variables

• The graph of a linear equation in two variables is always a straight line.
• To draw the graph:
  1. Find at least two solutions of the equation.
  2. Plot the points corresponding to these solutions on the coordinate plane.
  3. Join the points with a straight line.

Example: For the equation $x+y=4$:

1. Find two solutions: $(0,4)$ and $(4,0)$.
2. Plot the points $(0,4)$ and $(4,0)$ on the graph.
3. Join the points to get the straight line.

Steps to plot the graph:

1. Convert the equation into the form $y=mx+c$ (slope-intercept form), if necessary.
2. Identify the $y$-intercept $c$ (the point where the line crosses the Y-axis).
3. Use the slope $m$ to find another point.
4. Draw the line through these points.

4.4 Equation of a Line Parallel to Axes

• Equation of a line parallel to the X-axis:

  • If a line is parallel to the X-axis, the equation is of the form $y=k$, where $k$ is a constant.
  • Example: $y=3$ is a line parallel to the X-axis, passing through all points with ordinate 3.

• Equation of a line parallel to the Y-axis:

  • If a line is parallel to the Y-axis, the equation is of the form $x=k$, where $k$ is a constant.
  • Example: $x=4$ is a line parallel to the Y-axis, passing through all points with abscissa 4.

4.5 Intercepts

• The X-intercept is the point where the line crosses the X-axis, i.e., where $y=0$.
• The Y-intercept is the point where the line crosses the Y-axis, i.e., where $x=0$.

Example: For the equation $2x+3y=6$:

1. X-intercept: Set $y=0$ and solve for $x$:
   • $2x=6$, so $x=3$.
   • The X-intercept is $(3,0)$.
2. Y-intercept: Set $x=0$ and solve for $y$:
   • $3y=6$, so $y=2$.
   • The Y-intercept is $(0,2)$.

4.6 Equations Reducible to a Linear Form

Sometimes, complex equations can be simplified to a linear equation in two variables by rearranging or transforming terms.

Example: The equation $\frac{1}{x}+\frac{1}{y}=1$ can be transformed by substitution:

• Let $u=\frac{1}{x}$ and $v=\frac{1}{y}$.
• The equation becomes $u+v=1$, which is a linear equation.

Key Concepts

1. Linear Equation in Two Variables: An equation of the form $ax+by+c=0$.
2. Solution of a Linear Equation: A pair of values $(x,y)$ that satisfies the equation.
3. Graph of a Linear Equation: A straight line representing all the solutions of the equation.
4. X-intercept: The value of $x$ when $y=0$.
5. Y-intercept: The value of $y$ when $x=0$.
6. Equation of a Line Parallel to the X-axis: $y=k$.
7. Equation of a Line Parallel to the Y-axis: $x=k$.

Key Formulas

1. General Form of a linear equation: $ax+by+c=0$.
2. Slope-Intercept Form: $y=mx+c$, where $m$ is the slope and $c$ is the Y-intercept.

Practice Problems

1. Plot the graph of $2x+3y=6$.
2. Find the X- and Y-intercepts of the equation $x-2y=4$.
3. Write the equation of a line parallel to the X-axis passing through $(0,-3)$.
4. Solve the following pair of equations graphically:
   • $x+y=4$
   • $2x-y=1$