Pair of Linear Equations in Two Variables Class 10 Mathematics Concept notes and Formula Sheet

 

Concepts:

1. Introduction:

   • A linear equation in two variables is an equation of the form $ax + by = c$, where $a$, $b$, and $c$ are constants and $x$ and $y$ are the variables.
   • A pair of linear equations consists of two such equations.

2. Types of Solutions:

   • Consistent System: A system of equations has at least one solution. It can be either:
     • Unique Solution: The two lines intersect at exactly one point.
     • Infinitely Many Solutions: The two lines overlap, i.e., they are the same line.
   • Inconsistent System: The system has no solutions, meaning the two lines are parallel and never intersect.

3. Methods of Solving Pair of Linear Equations:

   • Graphical Method:
     • Plot both equations on a graph. The point(s) of intersection represent the solution(s).
   • Substitution Method:
     • Solve one equation for one variable and substitute this value into the other equation.
   • Elimination Method:
     • Add or subtract the equations to eliminate one variable, then solve for the remaining variable.
   • Matrix Method (Using Determinants):
     • Use matrices and determinants to solve the system. This method involves writing the equations in matrix form and applying Cramer's rule.

4. Graphical Method:

   • To solve the equations $ax + by = c$ and $dx + ey = f$:
     • Convert each equation to the slope-intercept form ($y = mx + c$).
     • Plot the lines and find the intersection point, which is the solution.

5. Substitution Method:

   • Solve one of the equations for one variable in terms of the other.
   • Substitute this expression into the other equation and solve for the remaining variable.
   • Substitute back to find the value of the first variable.

6. Elimination Method:

   • Multiply one or both equations to make the coefficients of one variable the same (or opposite).
   • Add or subtract the equations to eliminate one variable.
   • Solve the resulting equation for the remaining variable and substitute back to find the other variable.

7. Matrix Method:

   • For a system of equations: $$\begin{cases} a_1x + b_1y = c_1 \\ a_2x + b_2y = c_2 \end{cases}$$
     • Write in matrix form as $AX = B$, where: $$A = \begin{pmatrix} a_1 & b_1 \\ a_2 & b_2 \end{pmatrix}, \quad X = \begin{pmatrix} x \\ y \end{pmatrix}, \quad B = \begin{pmatrix} c_1 \\ c_2 \end{pmatrix}$$
     • Find the inverse of $A$ if it exists, and solve $X = A^{-1}B$.

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Formula Sheet:

1. Substitution Method:

   • Given equations: $$\begin{cases} a_1x + b_1y = c_1 \\ a_2x + b_2y = c_2 \end{cases}$$
   • Solve one equation for $x$ or $y$, substitute into the other equation, and solve.

2. Elimination Method:

   • Given equations: $$\begin{cases} a_1x + b_1y = c_1 \\ a_2x + b_2y = c_2 \end{cases}$$
   • Multiply the equations to align coefficients and then add or subtract to eliminate one variable.

3. Graphical Method:

   • Convert each equation to the form $y = mx + c$.
   • Plot the lines on a graph to find the intersection point.

4. Matrix Method:

   • Write the system of equations in matrix form $AX = B$: $$A = \begin{pmatrix} a_1 & b_1 \\ a_2 & b_2 \end{pmatrix}$$ $$X = \begin{pmatrix} x \\ y \end{pmatrix}$$ $$B = \begin{pmatrix} c_1 \\ c_2 \end{pmatrix}$$
   • Compute the inverse $A^{-1}$ and solve: $$X = A^{-1}B$$

5. Consistency Conditions:

   • Unique Solution: The lines intersect at exactly one point.
   • Infinitely Many Solutions: The lines are coincident (overlap).
   • No Solution: The lines are parallel (inconsistent system).