Linear Equations

Concept: The conditions for the number of solutions of a pair of linear equations in two variables, and , are determined by the ratios of their corresponding coefficients. Conditions for Solutions: (A) Unique Solution (Intersecting Lines): The pair of linear equations has exactly one unique solution if the lines they represent intersect at a single point. This occurs when the ratio of the coefficients of is not equal to the ratio of the coefficients of . Condition: (B) Infinitely Many Solutions (Coincident Lines): The pair of linear equations has infinitely many solutions if the lines they represent are coincident (i.e., they are the same line). This occurs when the ratios of all corresponding coefficients are equal. Condition: (C) No Solution (Parallel Distinct Lines): The pair of linear equations has no solution if the lines they represent are parallel and distinct. This occurs when the ratio of the coefficients of is equal to the ratio of the coefficients of , but this common ratio is not equal to the ratio of the constant terms. Condition: Step 1: Identify the condition for a unique solution Based on the concepts above, a pair of linear equations has a unique solution if and only if: Step 2: Compare with the given options
Option (1) : This is the condition for infinitely many solutions.
Option (2) : This condition alone is not sufficient to guarantee a unique solution. For example, if but (parallel lines), then would be true, but there's no solution.
Option (3) : This is the correct condition for a unique solution.
Option (4) : Similar to option (2), this condition alone is not sufficient. Therefore, the correct condition for a unique solution is .

Concept: For a pair of linear equations in two variables, and , the nature of their graphs (and solutions) can be determined by comparing the ratios of their coefficients: (A) If , the lines are intersecting (at exactly one point). (B) If , the lines are coincident (they are the same line). (C) If , the lines are parallel and distinct (no intersection). For perpendicular lines, the product of their slopes is -1 ( ). The slope . So, . Step 1: Identify the coefficients for both equations Equation 1: , , Equation 2: , , Step 2: Calculate the ratios and Step 3: Compare these two ratios We see that . So, . Step 4: Determine the nature of the lines Since , the lines are intersecting lines. They will meet at exactly one point. Step 5: Check for perpendicularity (optional, as intersecting is already an option) For lines to be perpendicular, . . Since , the lines are not perpendicular. The lines are intersecting (but not perpendicular). This matches option (4).

Concept: For a pair of linear equations in two variables, and , the nature of their solutions can be determined by comparing the ratios of their coefficients: (A) If , the lines are intersecting and there is exactly one unique solution. (B) If , the lines are coincident (they are the same line), and there are infinitely many solutions. (C) If , the lines are parallel and distinct, and there is no solution. Step 1: Write the given equations in the standard form Equation 1: Here, , , . Equation 2: Rewrite in standard form: Here, , , . Step 2: Calculate the ratios of the coefficients


Step 3: Compare the ratios We observe that: So, . Step 4: Determine the number of solutions Since all three ratios are equal ( ), the two linear equations represent coincident lines. This means they are essentially the same line, and every point on that line is a solution. Therefore, there are infinitely many solutions. This matches option (2). (Note: If you multiply the first equation by 2, you get , which is . This is identical to the second equation , confirming they are the same line.)

Concept: A line passes through the origin if the coordinates of the origin, , satisfy the equation of the line. This means if we substitute and into the equation, the equation should hold true (e.g., result in ). Alternatively, an equation of a line in the form passes through the origin if and only if the constant term is zero. Step 1: Understanding the condition for a line to pass through the origin The origin has coordinates . If a line passes through the origin, then substituting and into its equation must result in a true statement. Step 2: Check each option by substituting and
Option (1): Substitute : . This is false. So, this line does not pass through the origin. (This line is , a vertical line).
Option (2): Substitute : . This is false. So, this line does not pass through the origin. (This line is , a horizontal line).
Option (3): Substitute and : . This is false. So, this line does not pass through the origin.
Option (4): Substitute and : . This is true. So, this line passes through the origin. Step 3: Conclusion The equation represents a line passing through the origin because its constant term is zero, and substituting satisfies the equation.