Class 10 Maths Pair of Linear Equations in Two Variables Notes

3.1 Introduction to Pair of Linear Equations in Two Variables

A linear equation in two variables is an equation that can be written in the form:$$Ax + By + C = 0$$Ax+By+C=0

Where $A$A, $B$B, and $C$C are constants, and $x$x and $y$y are the variables. A pair of linear equations in two variables refers to two such equations, and we are required to find the values of $x$x and $y$y that satisfy both equations simultaneously.

For example:

1. $3x + 4y = 5$3x+4y=5
2. $2x – y = 1$2x−y=1

These are two linear equations, and our goal is to find the values of $x$x and $y$y that make both equations true at the same time.

The solution of a pair of linear equations can be found using different methods, including graphical and algebraic methods.

---

3.2 Graphical Method of Solution of a Pair of Linear Equations

The graphical method is one of the most visual ways to solve a pair of linear equations. To graphically solve the system of equations, follow these steps:

1. Convert both equations to the form y=mx+cy = mx + cy=mx+c or find intercepts:
   Rewrite each equation in slope-intercept form: $y = mx + c$y=mx+c, where $m$m is the slope and $c$c is the y-intercept.
2. Plot the lines on a graph:
   Plot both lines on the same graph, using the slope and intercept for each equation.
3. Identify the point of intersection:
   The point where the two lines intersect represents the values of $x$x and $y$y that satisfy both equations. This point is the solution.

• If the lines intersect at one point, the system has a unique solution (i.e., one pair of values for $x$x and $y$y).
• If the lines are parallel and never intersect, the system has no solution.
• If the lines overlap (coincide), the system has infinitely many solutions.

Example:

For the pair of equations:

1. $3x + 4y = 5$3x+4y=5
2. $2x – y = 1$2x−y=1

You would graph both equations, and where they intersect, that would be the solution.

---

3.3 Algebraic Methods of Solving a Pair of Linear Equations

The algebraic methods for solving a pair of linear equations are more efficient for exact solutions and are generally preferred in exams. These methods involve substitution and elimination, and we’ll explain each in detail.

---

3.3.1 Substitution Method

The substitution method involves solving one equation for one variable and then substituting this expression into the other equation.

Steps:

1. Solve one equation for one variable, say $x$x or $y$y.
2. Substitute the expression for $x$x or $y$y into the other equation.
3. Solve the resulting equation to find the value of one variable.
4. Substitute the value of the variable back into the original equation to find the value of the other variable.

Example:

Consider the system of equations:

1. $2x + 3y = 7$2x+3y=7
2. $x – y = 1$x−y=1

• Solve the second equation for $x$x: $$x = y + 1$$x=y+1
• Substitute $x = y + 1$x=y+1 into the first equation: $$2(y + 1) + 3y = 7$$2(y+1)+3y=7 $$2y + 2 + 3y = 7$$2y+2+3y=7 $$5y + 2 = 7$$5y+2=7 $$5y = 5 \quad \Rightarrow \quad y = 1$$5y=5⇒y=1
• Substitute $y = 1$y=1 into $x = y + 1$x=y+1: $$x = 1 + 1 = 2$$x=1+1=2

Thus, the solution is $x = 2$x=2 and $y = 1$y=1.

---

3.3.2 Elimination Method

The elimination method involves eliminating one variable by adding or subtracting the equations in such a way that one of the variables cancels out.

Steps:

1. Multiply both equations, if necessary, to make the coefficients of one variable the same.
2. Add or subtract the equations to eliminate one variable.
3. Solve the resulting equation for the remaining variable.
4. Substitute the value of the remaining variable into one of the original equations to find the other variable.

Example:

Consider the system of equations:

1. $3x + 2y = 16$3x+2y=16
2. $2x – y = 3$2x−y=3

• Multiply the second equation by 2 to make the coefficients of $y$y the same: $$4x – 2y = 6$$4x−2y=6
• Now, add the two equations: $$(3x + 2y) + (4x – 2y) = 16 + 6$$(3x+2y)+(4x−2y)=16+6 $$7x = 22 \quad \Rightarrow \quad x = \frac{22}{7}$$7x=22⇒x=722​
• Substitute $x = \frac{22}{7}$x=722​ into one of the original equations to find $y$y: $$3\left( \frac{22}{7} \right) + 2y = 16$$3(722​)+2y=16 $$\frac{66}{7} + 2y = 16$$766​+2y=16 $$2y = 16 – \frac{66}{7} = \frac{112}{7} – \frac{66}{7} = \frac{46}{7}$$2y=16−766​=7112​−766​=746​ $$y = \frac{23}{7}$$y=723​

Thus, the solution is $x = \frac{22}{7}$x=722​ and $y = \frac{23}{7}$y=723​.

---

3.4 Summary

In this chapter, we have learned about pairs of linear equations in two variables and various methods to solve them:

• Graphical Method: Visualizes the solution by graphing the equations and finding the point of intersection.
• Algebraic Methods: Involves solving the system algebraically using the substitution or elimination method.
  • Substitution Method: Solving one equation for one variable and substituting it into the other equation.
  • Elimination Method: Adding or subtracting equations to eliminate one variable.

MCQs Based on the “Pair of Linear Equations” Chapter:

1. Which of the following is a method of solving a pair of linear equations?

a) Substitution Method
b) Elimination Method
c) Graphical Method
d) All of the above

Answer: d) All of the above

2. In the graphical method of solving linear equations, the solution represents the point where the lines:

a) Parallel
b) Intersect
c) Perpendicular
d) Coincide

Answer: b) Intersect

3. The substitution method is used to solve a pair of linear equations by:

a) Graphing the equations
b) Adding or subtracting the equations
c) Substituting the value of one variable into the other equation
d) None of the above

Answer: c) Substituting the value of one variable into the other equation

4. If a pair of linear equations has infinitely many solutions, the two lines are:

a) Parallel
b) Perpendicular
c) Coincident
d) None of the above

Answer: c) Coincident