Pair of Linear Equations - Class 10 JKBOSE Mathematics

Pair of Linear Equations in Two Variables

Meaning and Structure of Linear Equations in Two Variables

A linear equation in two variables is an algebraic equation involving two unknowns, typically $x$ and $y$, with each variable having the highest power of one. The equation contains an equality sign and both variables must appear with non-zero coefficients. The general form is $ax + by + c = 0$, where $a$ and $b$ are not both zero.

• Variable: An unknown quantity whose value is not known.
• Equation: Any algebraic expression with an equality sign.
• Linear: The highest…

Pair of Linear Equations in Two Variables

Meaning and Structure of Linear Equations in Two Variables

A linear equation in two variables is an algebraic equation involving two unknowns, typically $x$ and $y$, with each variable having the highest power of one. The equation contains an equality sign and both variables must appear with non-zero coefficients. The general form is $ax + by + c = 0$, where $a$ and $b$ are not both zero.

• Variable: An unknown quantity whose value is not known.
• Equation: Any algebraic expression with an equality sign.
• Linear: The highest power of each variable is one.
• General Form: $ax + by + c = 0$, with $a, b \neq 0$.

Identifying Linear Equations in Two Variables

To determine if an equation is a linear equation in two variables, check for two variables, the presence of an equality sign, and that the power of each variable is one. If any variable has a power other than one, or if there is only one variable, it is not a linear equation in two variables.

• Example: $x + y = 1$ is a linear equation in two variables.
• Example: $2x + \sqrt{3}y = 0$ is also a linear equation in two variables.
• Example: $x^2 + y = 0$ is not, as $x$ has power two.

Solution of a Linear Equation in Two Variables

A solution to a linear equation in two variables is a pair of values $(x, y)$ that satisfy the equation. Substitute the values into the equation; if both sides are equal, the pair is a solution.

• For $x + y = 1$, $(0,1)$ and $(1,0)$ are solutions.
• If substituting values does not satisfy the equation, it is not a solution.

Pair of Linear Equations and Their Solutions

A pair of linear equations in two variables consists of two equations, each in the form $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$. A solution to the pair is a set of values $(x, y)$ that satisfy both equations simultaneously.

• The solution is any pair $(x, y)$ that makes both equations true.
• To check, substitute the values into both equations and verify equality.

Geometric Interpretation of Pair of Linear Equations

Each linear equation in two variables represents a straight line on the Cartesian plane. The solution to a pair of equations corresponds to the intersection point(s) of their lines.

• If the lines intersect at one point, there is a unique solution.
• If the lines are parallel, there is no solution.
• If the lines coincide, there are infinitely many solutions (every point on the line is a solution).

Algebraic Conditions for Number of Solutions

The relationship between the coefficients of the equations determines the number of solutions. The ratios of the coefficients are compared to classify the system.

• Unique Solution (Intersecting Lines): $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$
• No Solution (Parallel Lines): $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$
• Infinite Solutions (Coincident Lines): $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$

Consistent and Inconsistent Systems

A system is called consistent if it has at least one solution and inconsistent if it has none. The nature of the solution is determined by the above algebraic conditions.

• Consistent System: Has either a unique solution or infinitely many solutions.
• Inconsistent System: Has no solution.

Methods of Solving Pair of Linear Equations

There are two main methods for solving a pair of linear equations: graphical and algebraic. The algebraic method is further divided into substitution and elimination methods.

• Graphical Method: Plot both equations as lines and find their intersection.
• Algebraic Methods:
  • Substitution Method: Solve one equation for one variable and substitute into the other.
  • Elimination Method: Make the coefficients of one variable equal, then add or subtract equations to eliminate that variable.

Substitution Method (Stepwise)

The substitution method involves expressing one variable in terms of the other using one equation, then substituting this expression into the second equation.

1. Choose one equation and solve for one variable in terms of the other.
2. Substitute this expression into the other equation.
3. Solve for the remaining variable.
4. Substitute back to find the first variable.

Example: Given $2x + 3y = 13$ and $4x + 5y = 23$:

• From the second, $x = \frac{23 - 5y}{4}$.
• Substitute into the first: $2\left(\frac{23 - 5y}{4}\right) + 3y = 13$.
• Solve for $y$, then back-substitute to find $x$.

Elimination Method (Stepwise)

The elimination method involves manipulating the equations to eliminate one variable, allowing the other to be solved directly.

1. Multiply one or both equations to make the coefficients of one variable equal.
2. Add or subtract the equations to eliminate that variable.
3. Solve for the remaining variable.
4. Substitute back to find the other variable.

Example: Given $x + y = 3$ and $2x + 5y = 12$:

• Multiply the first by 2: $2x + 2y = 6$.
• Subtract from the second: $(2x + 5y) - (2x + 2y) = 12 - 6$.
• $3y = 6 \implies y = 2$, then $x = 1$.

Application Problems and Word Problems

Linear equations in two variables are used to model and solve real-life problems involving relationships between two unknowns. The process involves forming equations based on the problem statement and solving them using the above methods.

• Assign variables to unknowns.
• Translate conditions into equations.
• Use substitution or elimination to solve.
• Interpret the solution in the context of the problem.

Examples from transcript:

• Number of boys and girls in a quiz: $x + y = 10$, $y = x + 4$.
• Cost of pencils and pens: $5x + 7y = 50$, $7x + 5y = 46$.
• Taxi fare problem: $x + 10y = 105$, $x + 15y = 155$.
• Age problems, mixture problems, and digit problems are also modeled similarly.

Special Cases and Exam-Focused Takeaways

Certain cases require careful attention to the algebraic conditions and the nature of the equations.

• If the ratios of coefficients are equal except for the constants, the system is inconsistent (no solution).
• If all ratios are equal, the system has infinitely many solutions.
• Always check the consistency before attempting to solve.
• For word problems, define variables clearly and write equations systematically.
• Avoid random or unstructured solution steps; use proper mathematical formatting for full marks.

Table: Algebraic Condition and Number of Solutions

• For maximum marks, always write stepwise solutions and avoid random writing.
• Check the solution by substituting back into both equations.
• For graphical solutions, the intersection point gives the solution; if lines are parallel, there is no solution; if coincident, infinite solutions exist.