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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
Solution :
Given: A pair of linear equations in two variables:
Equation 1: $\frac{4}{3}x + 2y = 8$
Equation 2: $2x + 3y = 12$
To Find: Determine whether the given pair of linear equations is consistent or inconsistent by comparing the ratios $\frac{a_1}{a_2}$, $\frac{b_1}{b_2}$, and $\frac{c_1}{c_2}$.
Step 1: Standardizing the Equations
The standard form of a linear equation in two variables is $ax + by + c = 0$ or $ax + by = c$. We will express both equations in the form $ax + by = c$.
Equation 1: $\frac{4}{3}x + 2y = 8$
Equation 2: $2x + 3y = 12$
Step 2: Identifying Coefficients
Comparing the given equations with the general form $a_1x + b_1y = c_1$ and $a_2x + b_2y = c_2$:
For Equation 1: $a_1 = \frac{4}{3}$, $b_1 = 2$, $c_1 = 8$
For Equation 2: $a_2 = 2$, $b_2 = 3$, $c_2 = 12$
Step 3: Calculating the Ratios
Now, we calculate the ratios of the coefficients:
Ratio of $x$-coefficients: $\frac{a_1}{a_2} = \frac{4/3}{2} = \frac{4}{3 \times 2} = \frac{4}{6} = \frac{2}{3}$
Ratio of $y$-coefficients: $\frac{b_1}{b_2} = \frac{2}{3}$
Ratio of constants: $\frac{c_1}{c_2} = \frac{8}{12} = \frac{8 \div 4}{12 \div 4} = \frac{2}{3}$
Step 4: Comparing the Ratios and Determining Consistency
We observe that:
$\frac{a_1}{a_2} = \frac{2}{3}$
$\frac{b_1}{b_2} = \frac{2}{3}$
$\frac{c_1}{c_2} = \frac{2}{3}$
Therefore, $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$.
[Justification: According to the algebraic condition for consistency, if $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$, the lines are coincident, meaning they have infinitely many solutions. A system of linear equations is defined as consistent if it has at least one solution.]
Since the ratios are equal, the pair of linear equations represents coincident lines and possesses infinitely many solutions.
Final Answer: The pair of linear equations is consistent.
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
- 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(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
- 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.
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