Kaliyuga Ekalavya

NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Question: On comparing the ratio, \( \left(\frac{a_1}{a_2}\right), \left(\frac{b_1}{b_2}\right), \left(\frac{c_1}{c_2}\right) \), find out whether the following pair of linear equations are consistent, or inconsistent. \( \frac{4}{3}x + 2y = 8 \) ; \( 2x + 3y = 12 \) Given: The two linear equations are: (1) \( \frac{4}{3}x + 2y = 8 \) (2) \( 2x + 3y = 12 \) To Find: Whether the given pair of linear equations are consistent or inconsistent by comparing the ratios: \( \left(\frac{a_1}{a_2}\right), \left(\frac{b_1}{b_2}\right), \left(\frac{c_1}{c_2}\right) \) Formula: For two linear equations in the form: \( a_1x + b_1y + c_1 = 0 \) \( a_2x + b_2y + c_2 = 0 \) The consistency is checked as follows: Case 1: If \( \frac{a_1}{a_2} \ne \frac{b_1}{b_2} \), equations are consistent and have a unique solution. Case 2: If \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \ne \frac{c_1}{c_2} \), equations a...

NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Question: On comparing the ratios a 1 , b 1 , c 1 and a 2 , b 2 , c 2 , find out whether the following pair of linear equations are consistent or inconsistent: \(2x - 3y = 8;\quad 4x - 6y = 9.\) Given: \(2x - 3y = 8\) \(4x - 6y = 9\) To Find: Whether the given pair of linear equations is consistent or inconsistent by comparing the ratios of coefficients. Formula: For a pair of linear equations: \(a_1x + b_1y + c_1 = 0\) \(a_2x + b_2y + c_2 = 0\) Consistent & Unique Solution: if \(\dfrac{a_1}{a_2} \neq \dfrac{b_1}{b_2}\) Consistent & Infinitely Many Solutions: if \(\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}\) Inconsistent: if \(\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \neq \dfrac{c_1}{c_2}\) Solution: Step 1: Write both equations in standard form. $$ \begin{align*} 2x - 3y &= 8 \implies 2x - 3y - 8 = 0 \\ 4x - 6...

NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Exercise 3.1 Question 3(iv) Question: On comparing the ratios a 1 , b 1 , c 1 and a 2 , b 2 , c 2 , find out whether the following pair of linear equations are consistent or inconsistent. \( 5x - 3y = 11 \) ; \( -10x + 6y = -22 \) Given: The pair of linear equations: 1) \( 5x - 3y = 11 \) 2) \( -10x + 6y = -22 \) To Find: Whether the given pair of equations is consistent or inconsistent by comparing the ratios: \[ \frac{a_1}{a_2},\quad \frac{b_1}{b_2},\quad \frac{c_1}{c_2} \] Formula: For two linear equations in the form: \( a_1x + b_1y + c_1 = 0 \) \( a_2x + b_2y + c_2 = 0 \) Compare the ratios: \[ \frac{a_1}{a_2},\quad \frac{b_1}{b_2},\quad \frac{c_1}{c_2} \] Cases: Case 1: If \( \frac{a_1}{a_2} \ne \frac{b_1}{b_2} \), the system is consistent (has a unique solution). Case 2: If \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \ne \frac{c_1}{c_2} \), the system is inconsist...

NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Exercise 3.1 Question 4(i) Question: Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically: \( x + y = 5 \), \( 2x + 2y = 10 \) Given: The pair of linear equations: 1) \( x + y = 5 \) 2) \( 2x + 2y = 10 \) To Find: Whether the given pair of equations is consistent or inconsistent. If consistent, obtain the solution graphically. Formula: For two linear equations \( a_1x + b_1y = c_1 \) and \( a_2x + b_2y = c_2 \): If \( \dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2} \), the equations are dependent and consistent (infinitely many solutions). If \( \dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \neq \dfrac{c_1}{c_2} \), the equations are inconsistent (no solution). If \( \dfrac{a_1}{a_2} \neq \dfrac{b_1}{b_2} \), the equations are consistent and independent (unique solution). Graphic...

NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Exercise 3.1 Question 3(iii) Question: On comparing the ratios a_1 , b_1 , c_1 and a_2 , b_2 , c_2 , find out whether the following pair of linear equations are consistent or inconsistent: \[ \frac{3}{2}x + \frac{5}{3}y = 7; \] \[ 9x - 10y = 14. \] Given: The two linear equations are: (1) \( \frac{3}{2}x + \frac{5}{3}y = 7 \) (2) \( 9x - 10y = 14 \) To Find: Whether the given pair of linear equations is consistent or inconsistent by comparing the ratios: \[ \frac{a_1}{a_2}, \quad \frac{b_1}{b_2}, \quad \frac{c_1}{c_2} \] Formula: For a pair of linear equations: \[ a_1x + b_1y + c_1 = 0 \] \[ a_2x + b_2y + c_2 = 0 \] If \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \), the equations are consistent and have a unique solution. If \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \), the equations are inconsistent (no solution). If \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}...

NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Question: Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically: \( x - y = 8 \), \( 3x - 3y = 16 \) Given: The given pair of linear equations is: 1) \( x - y = 8 \) 2) \( 3x - 3y = 16 \) To Find: Whether the given pair of equations is consistent or inconsistent. If consistent, obtain the solution graphically. Formula: Two linear equations in two variables are consistent if they have at least one solution (i.e., they intersect or are coincident). For equations in the form \( a_1x + b_1y + c_1 = 0 \) and \( a_2x + b_2y + c_2 = 0 \): Consistent if \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \), or \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \) (Inconsistent if all three ratios are equal). To solve graphically, plot both equations and find their point of intersection. Solution: Step 1: Write both eq...

NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Exercise 3.1 Question 4(iii) Question: Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically: \( 2x + y - 6 = 0 \), \( 4x - 2y - 4 = 0 \) Given: Equation 1: \( 2x + y - 6 = 0 \) Equation 2: \( 4x - 2y - 4 = 0 \) To Find: Whether the given pair of equations is consistent or inconsistent. If consistent, find the solution graphically. Formula: For two linear equations in the form \( a_1x + b_1y + c_1 = 0 \) and \( a_2x + b_2y + c_2 = 0 \): If \( \dfrac{a_1}{a_2} \neq \dfrac{b_1}{b_2} \), the system is consistent and has a unique solution. If \( \dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \neq \dfrac{c_1}{c_2} \), the system is inconsistent (no solution). If \( \dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2} \), the system has infinitely many solutions (consistent). Solution: Step 1: Write the equa...

NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Question: Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically: \(2x - 2y - 2 = 0\), \(4x - 4y - 5 = 0\) Given: Pair of linear equations: 1) \(2x - 2y - 2 = 0\) 2) \(4x - 4y - 5 = 0\) To Find: Are the given pair of equations consistent or inconsistent? If consistent, obtain the solution graphically. Formula: General form of a linear equation: \(ax + by + c = 0\) To check consistency for equations \(a_1x + b_1y + c_1 = 0\) and \(a_2x + b_2y + c_2 = 0\): If \(\dfrac{a_1}{a_2} \ne \dfrac{b_1}{b_2}\), the system is consistent and has a unique solution. If \(\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \ne \dfrac{c_1}{c_2}\), the system is inconsistent (no solution). If \(\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}\), the system is consistent with infinitely many solutions. Solution: Step 1: Write the equat...

NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Question: Half the perimeter of a rectangular garden, whose length is 4 m more than its width, is 36 m. Find the dimensions of the garden. Given: Let the width of the garden be \( x \) m. Length of the garden is \( x + 4 \) m. Half the perimeter is 36 m. To Find: The length and width of the garden. Formula: Perimeter of a rectangle \( = 2 \times (\text{Length} + \text{Width}) \) Half perimeter \( = \text{Length} + \text{Width} \) Solution: Step 1: Let the width of the garden be \( x \) m. Then, the length is \( x + 4 \) m. $$ \text{Width} = x \text{ m} \\ \text{Length} = x + 4 \text{ m} $$ Step 2: According to the question, half the perimeter is 36 m. So, $$ \text{Length} + \text{Width} = 36 \\ x + (x + 4) = 36 $$ Step 3: Simplify the equation. $$ x + x + 4 = 36 \\ 2x + 4 = 36 $$ Step 4: Subtract 4 from both sides. $$ 2x + 4 - 4 = 36 - 4 \\ 2x = 32 $$ ...

NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Question: Given the linear equation \(2x + 3y - 8 = 0\), write another linear equation in two variables such that the geometrical representation of the pair so formed is: (i) intersecting lines (ii) parallel lines (iii) coincident lines. Given: The first linear equation is: $$ 2x + 3y - 8 = 0 $$ To Find: Another linear equation in two variables such that the pair of equations represents: (i) Intersecting lines (ii) Parallel lines (iii) Coincident lines Formula: For two linear equations in the form: $$ \begin{aligned} a_1x + b_1y + c_1 &= 0 \\ a_2x + b_2y + c_2 &= 0 \end{aligned} $$ The nature of the pair is determined as follows: Intersecting lines: \(\dfrac{a_1}{a_2} \neq \dfrac{b_1}{b_2}\) Parallel lines: \(\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \neq \dfrac{c_1}{c_2}\) Coincident lines: \(\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} = \dfrac{c_1}{c_2}\) Solution: St...

NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Question: Draw the graphs of the equations \(x - y + 1 = 0\) and \(3x + 2y - 12 = 0\). Determine the coordinates of the vertices of the triangle formed by these lines and the x-axis, and shade the triangular region. Given: Equation 1: \(x - y + 1 = 0\) Equation 2: \(3x + 2y - 12 = 0\) The x-axis (\(y = 0\)) To Find: The coordinates of the vertices of the triangle formed by the two lines and the x-axis. Graphical representation with the triangular region shaded. Formula: To find intersection of two lines: Solve the two equations simultaneously. To find intersection with x-axis: Substitute \(y = 0\) in the equation of the line. Solution: Step 1: Find the x-intercept of \(x - y + 1 = 0\). Substitute \(y = 0\): \[ x - 0 + 1 = 0 \implies x + 1 = 0 \implies x = -1 \] So, the intersection with the x-axis is \((-1, 0)\). Step 2: Find the x-intercept of \(3x + 2y - 12 =...

NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Example 7 Question: Two rails are represented by the equations x + 2y – 4 = 0 and 2x + 4y – 12 = 0. Will the rails cross each other? Given: Equation of rail 1: x + 2y – 4 = 0 Equation of rail 2: 2x + 4y – 12 = 0 To Find: Whether the two rails cross each other. Formula: If two lines are represented by the equations a 1 x + b 1 y + c 1 = 0 and a 2 x + b 2 y + c 2 = 0, then the lines are parallel if a 1 a 2 = b 1 b 2 . Solution: Let the given equations be x + 2y – 4 = 0 .....(1) 2x + 4y – 12 = 0 .....(2) From equation (1), we have x = 4 - 2y Substituting this in equation (2): 2(4 - 2y) + 4y – 12 = 0 ⇒ 8 – 4y + 4y – 12 = 0 ⇒ –4 = 0 This is a contradiction. Alternatively, we can check if the lines are parallel: For equation (1): a 1 = 1, b 1 = 2 For equation (2): a 2 = 2, b 2 = 4 a 1 a 2 = 1 2 = 1 2 b 1 b 2 = 2 4 = 1 2 Since a 1 a 2 = b 1 b 2 , the lines are parallel...

NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Exercise 3.2 Question 1(ii) Question: Solve the following pair of linear equations by the substitution method: s – t = 3 and s 3 + t 2 = 6 Given: The given equations are: s – t = 3 ---(1) s 3 + t 2 = 6 ---(2) To Find: We need to find the values of s and t using the substitution method. Formula: Substitution method involves solving one equation for one variable and substituting it into the other equation. Solution: From equation (1), we have s = t + 3. Substitute s = t + 3 into equation (2): t+3 3 + t 2 = 6 Multiplying both sides by 6 to eliminate fractions: 2(t+3) + 3t = 36 ⇒ 2t + 6 + 3t = 36 ⇒ 5t = 30 ⇒ t = 6 Substitute t = 6 into s = t + 3: s = 6 + 3 = 9 Result: Therefore, the solution is s = 9 and t = 6. Next question solution: NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Exercise 3.2 Question 1(iii) Explore more in Pair of L...

NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Exercise 3.2 Question 1(iii) Question: Solve the following pair of linear equations by the substitution method: 3x – y = 3 9x – 3y = 9 Given: The given pair of linear equations is: 3x – y = 3 ---(1) 9x – 3y = 9 ---(2) To Find: We need to find the values of x and y that satisfy both equations using the substitution method. Formula: Substitution method involves solving one equation for one variable and substituting it into the other equation. Solution: From equation (1), we can express y in terms of x: y = 3x – 3 Substitute this expression for y into equation (2): 9x – 3(3x – 3) = 9 Simplify and solve for x: 9x – 9x + 9 = 9 9 = 9 This equation is always true, indicating that the two equations are linearly dependent (represent the same line). Therefore, there are infinitely many solutions. Any point (x, y) that satisfies equation (1) also satisfies equation (2). Result: Th...

NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Exercise 3.2 Question 1(i) Question: Solve the following pair of linear equations by the substitution method. x + y = 14 x – y = 4 Given: x + y = 14 ...(1) x – y = 4 ...(2) To Find: The values of x and y that satisfy both equations. Formula: Substitution method: Solve one equation for one variable, substitute into the other equation. Solution: From equation (2): x = y + 4 Substitute x = y + 4 into equation (1): (y + 4) + y = 14 ⇒ 2y + 4 = 14 ⇒ 2y = 10 ⇒ y = 5 Substitute y = 5 into x = y + 4: x = 5 + 4 ⇒ x = 9 Result: Therefore, x = 9 and y = 5 Next question solution: NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Exercise 3.2 Question 1(ii) Explore more in Pair of Linear Equations: Click this link to explore more NCERT Class X Chapter 3 Pair of Linear Equations Explore more: Click this link to explore more NCERT Class X chapter solut...

NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Exercise 3.2 Question 1(iv) Question: Solve the following pair of linear equations by the substitution method: 0.2x + 0.3y = 1.3 and 0.4x + 0.5y = 2.3 Given: The given equations are: 0.2x + 0.3y = 1.3 ---(1) 0.4x + 0.5y = 2.3 ---(2) To Find: We need to find the values of x and y that satisfy both equations. Formula: Substitution method: Solve one equation for one variable and substitute it into the other equation. Solution: From equation (1), we can express x in terms of y: 0.2x = 1.3 - 0.3y ⇒ x = 1.3 - 0.3y 0.2 Substitute this value of x into equation (2): 0.4( 1.3 - 0.3y 0.2 ) + 0.5y = 2.3 Simplify and solve for y: 2(1.3 - 0.3y) + 0.5y = 2.3 ⇒ 2.6 - 0.6y + 0.5y = 2.3 ⇒ -0.1y = -0.3 ⇒ y = 3 Substitute y = 3 back into the expression for x: x = 1.3 - 0.3(3) 0.2 = 1.3 - 0.9 0.2 = 0.4 0.2 = 2 Result: Therefore, the solution to the pair of linear equations is x = 2 and y...

NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Exercise 3.2 Question 1(v) Question: Solve the following pair of linear equations by the substitution method. √2x + √3y = 0 √3x - √8y = 0 Given: √2x + √3y = 0 √3x - √8y = 0 To Find: The values of x and y that satisfy both equations. Formula: Substitution method for solving linear equations. Solution: From the first equation, √2x + √3y = 0 ⇒ √2x = -√3y ⇒ x = -√3y √2 Substitute this value of x into the second equation: √3( -√3y √2 ) - √8y = 0 Simplifying, we get: -3y √2 - √8y = 0 ⇒ -3y √2 = 2√2y -3y = 4y ⇒ 7y = 0 ⇒ y = 0 Substitute y = 0 into x = -√3y √2 ⇒ x = 0 Result: x = 0, y = 0 Next question solution: NCERT Class X Chapter 3: Pair of Linear Equations In Two Variables Exercise 3.2 Question 1(vi) Explore more in Pair of Linear Equations: Click this link to explore more NCERT Class X Chapter 3 Pair of Linear Equations Explore more: Click this link to exp...