Ex.3.2 Q3 Pair of Linear Equations in Two Variables Solution - NCERT Maths Class 10

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Question

On comparing the ratios \(\begin{align}\frac{{{a_1}}}{{{a_2}}},\frac{{{b_1}}}{{{b_2}}},\frac{{{c_1}}}{{{c_2}}}\end{align}\), find out whether the following pair of linear equations are consistent, or inconsistent. 

(i) \(3x + 2y = 5;\;\;2x-3y = 7\)

(ii) \(2x-3y = 8;\;\;4x-6y = 9\)

(iii) \(\begin{align} \frac{3}{2}x + \frac{5}{3}y = 7;\;\;9x-10y = 14 \end{align}\)

(iv) \(5x-3y = 11;\;\;-10x + 6y = -22\)

(v) \(\begin{align} \frac{4}{3}x + 2y = 8;\;\;2x + 3y = 12 \end{align}\)

 Video Solution
Pair Of Linear Equations In Two Variables
Ex 3.2 | Question 3

Text Solution

What is Unknown?

To find out whether the linear equations are consistent or inconsistent.

Reasoning:

For any pair of linear equation

\[\begin{align}{a_1}x + {b_1}y + {c_1} &= 0\\{a_2}x + {b_2}y + {c_2} &= 0\end{align}\]

Consistent means pair of linear equations have one solution or infinitely many solutions.

\[\begin{align}
\frac{{{a_1}}}{{{a_2}}} \ne \frac{{{b_1}}}{{{b_2}}}\,\,\,\,\,\,\,\,\,\,\,\;\; & \;\left( {{\bf{Intersecting}}{\rm{\, }}{\bf{lines}}/{\bf{one}}\;{\bf{Solution}}} \right)\\
\,\frac{{{a_1}}}{{{a_2}}} = \frac{{{b_1}}}{{{b_2}}} = \frac{{{c_1}}}{{{c_2}}} & \,\;\left( {{\bf{Coincident}}{\rm{ \,}}{\bf{Lines}}/{\bf{Infinitely}}\;{\bf{many}}\;{\bf{Solutions}}} \right)
\end{align}\]

Inconsistent means, the lines may be parallel and do not have any Solution)

\[\frac{{{a_1}}}{{{a_2}}} = \frac{{{b_1}}}{{{b_2}}} \ne \,\,\frac{{{c_1}}}{{{c_2}}}\,\,\,\,\,\,\,\,\,\left( {{\bf{Parallel}}{\rm{\; }}{\bf{lines}}/{\bf{No}}\;{\bf{Solution}}} \right)\]

(i) What is Known?

\(\begin{align}3x + 2y - 5 &= 0{\rm{ }}\\2x-3y - 7 &= 0\end{align}\)

Steps:

\[\begin{align}\frac{{{a_1}}}{{{a_2}}} &= \frac{3}{2}\\\frac{{{b_1}}}{{{b_2}}} &= \frac{2}{{ - 3}}\\
\frac{{{c_1}}}{{{c_2}}} &= \frac{{ - 5}}{{ - 7}} \\&= \frac{5}{7}\end{align}\]

From above

\[\frac{{{a_1}}}{{{a_2}}} \ne \frac{{{b_1}}}{{{b_2}}}\]

Therefore, lines are intersecting and have one solution,

Hence, the pair of equations are consistent.

(ii) What is Known:

\(\begin{align}2x-3y - 8 &= 0\\4x-6y - 9 &= 0\end{align}\)

Steps:

\[\begin{align}\frac{{{a_1}}}{{{a_2}}} &= \frac{2}{4} = \frac{1}{2}\\\frac{{{b_1}}}{{{b_2}}} &= \frac{{ - 3}}{{ - 6}} = \frac{1}{2}\\\frac{{{c_1}}}{{{c_2}}} &= \frac{{ - 8}}{{ - 9}} = \frac{8}{9}\end{align}\]

From above

\[\frac{{{a_1}}}{{{a_2}}} = \frac{{{b_1}}}{{{b_2}}} \ne \,\,\frac{{{c_1}}}{{{c_2}}}\,\]

Therefore, lines are parallel and have no solution,

Hence, the pair of equations are inconsistent.

(iii) What is Known?

\(\begin{align}\frac{3}{2}x + \frac{5}{3}y &= 7{\rm{ }}\\9x - 10y &= 14\end{align}\)

Steps:

\[\begin{align}\frac{{{a_1}}}{{{a_2}}} &= \frac{{\frac{3}{2}}}{9} \\&= \frac{3}{2} \times \frac{1}{9}\\& = \frac{1}{6}\\
\frac{{{b_1}}}{{{b_2}}} &= \frac{{\frac{5}{3}}}{{ - 10}} \\&= \frac{5}{3} \times \frac{1}{{ - 10}}\\& = \frac{1}{{ - 6}}\\
\frac{{{c_1}}}{{{c_2}}} &= \frac{7}{{14}} \\&= \frac{1}{2}\end{align}\]

From above

\[\frac{{{a_1}}}{{{a_2}}} \ne \frac{{{b_1}}}{{{b_2}}}\,\]

Therefore, lines are intersecting and have one solution.

Hence, they are consistent.

(iv) What is Known?

\(\begin{align}5x-3y - 11 &= 0{\rm{ }}\\-10x + 6y + 22 &= 0\end{align}\)

Steps:

\[\begin{align}\frac{{{a_1}}}{{{a_2}}} &= \frac{5}{{ - 10}} \\&= \frac{{ - 1}}{2}\\
\frac{{{b_1}}}{{{b_2}}} &= \frac{{ - 3}}{6} \\&= \frac{{ - 1}}{2}\\\frac{{{c_1}}}{{{c_2}}}& = \frac{{ - 11}}{{22}} \\&= \frac{{ - 1}}{2}\end{align}\]

From above

\[\frac{{{a_1}}}{{{a_2}}} = \frac{{{b_1}}}{{{b_2}}} = \,\,\frac{{{c_1}}}{{{c_2}}}\,\]

Therefore, lines are coincident and have infinitely many solutions.

Hence, they are consistent.

(v) What is Known?

\(\begin{align}\frac{4}{3}x + 2y &= 8\\2x + 3y &= 12\end{align}\)

Steps:

\[\begin{align}\frac{{{a_1}}}{{{a_2}}}& = \frac{{\frac{4}{3}}}{2} = \frac{4}{3} \times \frac{1}{2} \\&= \frac{2}{3}\\
\frac{{{b_1}}}{{{b_2}}} &= \frac{2}{3}\\\frac{{{c_1}}}{{{c_2}}} &= \frac{{ - 8}}{{ - 12}}\\& = \frac{2}{3}\end{align}\]

From above

\[\frac{{{a_1}}}{{{a_2}}} = \frac{{{b_1}}}{{{b_2}}} = \,\,\frac{{{c_1}}}{{{c_2}}}\,\]

Therefore, lines are coincident and have infinitely many solutions.

Hence, they are consistent.