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Q4(ii):
Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically: (ii) x – y = 8, 3x – 3y = 16
Solution :
Given: A pair of linear equations in two variables:
Equation 1: $x - y = 8$
Equation 2: $3x - 3y = 16$
To Find: Determine whether the given pair of linear equations is consistent or inconsistent. If consistent, obtain the solution graphically.
Step 1: Standardizing the Equations
A pair of linear equations in the form $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$ can be analyzed by comparing the ratios of their coefficients.
Rewrite the equations in standard form:
Equation 1: $1x - 1y - 8 = 0$
Equation 2: $3x - 3y - 16 = 0$
Step 2: Comparing Ratios of Coefficients
Identify the coefficients:
$a_1 = 1, b_1 = -1, c_1 = -8$
$a_2 = 3, b_2 = -3, c_2 = -16$
Calculate the ratios:
$\frac{a_1}{a_2} = \frac{1}{3}$
$\frac{b_1}{b_2} = \frac{-1}{-3} = \frac{1}{3}$
$\frac{c_1}{c_2} = \frac{-8}{-16} = \frac{1}{2}$
Step 3: Analyzing Consistency
[Theorem: If $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$, the lines are parallel and the system has no solution (inconsistent).]
Since $\frac{1}{3} = \frac{1}{3} \neq \frac{1}{2}$, the condition for inconsistency is satisfied.
Visual Representation (Conceptual):
Step 4: Conclusion
Because the ratios of the coefficients of $x$ and $y$ are equal, but not equal to the ratio of the constant terms, the lines represented by these equations are parallel to each other. Parallel lines never intersect, meaning there is no common point that satisfies both equations simultaneously.
Final Answer: The pair of linear equations is inconsistent.
More Questions from Class 10 Mathematics Pair of linear equations in two variable EXERCISE 3.1
- Q1(i): Form the pair of linear equations in the following problems, and find their solutions graphically. (i) 10 students of Class X took part in a Mathematics quiz. If the number of girls is 4 more than the number of boys, find the number of boys and girls who took part in the quiz.
- Q1(ii): Form the pair of linear equations in the following problems, and find their solutions graphically. (ii) 5 pencils and 7 pens together cost ` 50, whereas 7 pencils and 5 pens together cost ` 46. Find the cost of one pencil and that of one pen.
- Q2(i): On comparing the ratios $\frac{a_1}{a_2}$, $\frac{b_1}{b_2}$, and $\frac{c_1}{c_2}$, find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident: (i) 5x – 4y + 8 = 0; 7x + 6y – 9 = 0
- Q2(ii): On comparing the ratios $\frac{a_1}{a_2}$, $\frac{b_1}{b_2}$, and $\frac{c_1}{c_2}$, find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident: (ii) 9x + 3y + 12 = 0; 18x + 6y + 24 = 0
- Q2(iii): On comparing the ratios $\frac{a_1}{a_2}$, $\frac{b_1}{b_2}$, and $\frac{c_1}{c_2}$, find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident: (iii) 6x – 3y + 10 = 0; 2x – y + 9 = 0
- Q3(i): On comparing the ratios $\frac{a_1}{a_2}$, $\frac{b_1}{b_2}$, and $\frac{c_1}{c_2}$, find out whether the following pair of linear equations are consistent, or inconsistent. (i) 3x + 2y = 5 ; 2x – 3y = 7
- Q3(ii): On comparing the ratios $\frac{a_1}{a_2}$, $\frac{b_1}{b_2}$, and $\frac{c_1}{c_2}$, find out whether the following pair of linear equations are consistent, or inconsistent. (ii) 2x – 3y = 8 ; 4x – 6y = 9
- Q3(iii): On comparing the ratios $\frac{a_1}{a_2}$, $\frac{b_1}{b_2}$, and $\frac{c_1}{c_2}$, find out whether the following pair of linear equations are consistent, or inconsistent. (iii) $\frac{3}{2}x + \frac{5}{3}y = 7$ ; 9x – 10y = 14
- Q3(iv): On comparing the ratios $\frac{a_1}{a_2}$, $\frac{b_1}{b_2}$, and $\frac{c_1}{c_2}$, find out whether the following pair of linear equations are consistent, or inconsistent. (iv) 5x – 3y = 11 ; – 10x + 6y = –22
- Q3(v): On comparing the ratios $\frac{a_1}{a_2}$, $\frac{b_1}{b_2}$, and $\frac{c_1}{c_2}$, find out whether the following pair of linear equations are consistent, or inconsistent. (v) $\frac{4}{3}x + 2y = 8$ ; 2x + 3y = 12
- Q4(i): Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically: (i) x + y = 5, 2x + 2y = 10
- Q4(iii): Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically: (iii) 2x + y – 6 = 0, 4x – 2y – 4 = 0
- Q4(iv): Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically: (iv) 2x – 2y – 2 = 0, 4x – 4y – 5 = 0
- Q5: 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.
- Q6(i): 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
- Q6(ii): 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: (ii) parallel lines
- Q6(iii): 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: (iii) coincident lines
- Q7: 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.
CBSE Solutions for Class 10 Mathematics Pair of linear equations in two variable
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