Answer

Verified

466.5k+ views

Hint: We first find the nature of system of linear equations, if the nature of system of equations are intersecting lines then we will have a unique solution, if the nature of system of equations are coincident then we will have infinite number of solutions, if the nature of system of equations are parallel then there will be no solutions.

Complete step-by-step answer:

First, we will find the nature of a pair of linear equations.

If ${{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}=0$ and ${{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}=0$ are pair of linear equations in two variables

If $\dfrac{{{a}_{1}}}{{{a}_{2}}}\ne \dfrac{{{b}_{1}}}{{{b}_{2}}}$ then lines are intersecting.

If $\dfrac{{{a}_{1}}}{{{a}_{2}}}=\dfrac{{{b}_{1}}}{{{b}_{2}}}\ne \dfrac{{{c}_{1}}}{{{c}_{2}}}$ then lines are parallel.

If $\dfrac{{{a}_{1}}}{{{a}_{2}}}=\dfrac{{{b}_{1}}}{{{b}_{2}}}=\dfrac{{{c}_{1}}}{{{c}_{2}}}$ then lines are coincident.

Pair of linear equations are $3=2x+y\cdot \cdot \cdot \cdot \cdot (1)$ and $9=4x-y\cdot \cdot \cdot \cdot \cdot (2)$

Now, we will find the nature of linear equations.

$\begin{align}

& {{a}_{1}}=2,{{b}_{1}}=1,{{c}_{1}}=-3 \\

& {{a}_{2}}=4,{{b}_{2}}=-1,{{c}_{2}}=9 \\

\end{align}$$$$$

$\dfrac{{{a}_{1}}}{{{a}_{2}}}=\dfrac{2}{4}=\dfrac{1}{2}$ and $\dfrac{{{b}_{1}}}{{{b}_{2}}}=\dfrac{1}{-1}=-1$

Here $\dfrac{{{a}_{1}}}{{{a}_{2}}}\ne \dfrac{{{b}_{1}}}{{{b}_{2}}}$ then lines are intersecting.

So, it has a unique solution.

So, now we will solve for x and y.

We will add both equations (1) and (2)

$\Rightarrow 12=6x$

$\Rightarrow x=2$

Now we will substitute the obtained value of $x$ in either equation 1 or in equation 2. Now we will substitute value of $x$ i.e. 2 in equation 1 we will get,

$\begin{align}

& \Rightarrow 3=2\times (2)+y \\

& \Rightarrow y=3-4 \\

& \Rightarrow y=-1 \\

\end{align}$

We obtained the value of $x$ and $y$ by solving equation 1 and equation 2.

The values of $x$ and $y$ are 2 and -1 respectively.

$x=2,y=-1$.

Note: While solving a pair of linear equations in two variables first we will find the nature of a pair of linear equations in two variables and then, we proceed to solve if lines are not parallel.

Complete step-by-step answer:

First, we will find the nature of a pair of linear equations.

If ${{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}=0$ and ${{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}=0$ are pair of linear equations in two variables

If $\dfrac{{{a}_{1}}}{{{a}_{2}}}\ne \dfrac{{{b}_{1}}}{{{b}_{2}}}$ then lines are intersecting.

If $\dfrac{{{a}_{1}}}{{{a}_{2}}}=\dfrac{{{b}_{1}}}{{{b}_{2}}}\ne \dfrac{{{c}_{1}}}{{{c}_{2}}}$ then lines are parallel.

If $\dfrac{{{a}_{1}}}{{{a}_{2}}}=\dfrac{{{b}_{1}}}{{{b}_{2}}}=\dfrac{{{c}_{1}}}{{{c}_{2}}}$ then lines are coincident.

Pair of linear equations are $3=2x+y\cdot \cdot \cdot \cdot \cdot (1)$ and $9=4x-y\cdot \cdot \cdot \cdot \cdot (2)$

Now, we will find the nature of linear equations.

$\begin{align}

& {{a}_{1}}=2,{{b}_{1}}=1,{{c}_{1}}=-3 \\

& {{a}_{2}}=4,{{b}_{2}}=-1,{{c}_{2}}=9 \\

\end{align}$$$$$

$\dfrac{{{a}_{1}}}{{{a}_{2}}}=\dfrac{2}{4}=\dfrac{1}{2}$ and $\dfrac{{{b}_{1}}}{{{b}_{2}}}=\dfrac{1}{-1}=-1$

Here $\dfrac{{{a}_{1}}}{{{a}_{2}}}\ne \dfrac{{{b}_{1}}}{{{b}_{2}}}$ then lines are intersecting.

So, it has a unique solution.

So, now we will solve for x and y.

We will add both equations (1) and (2)

$\Rightarrow 12=6x$

$\Rightarrow x=2$

Now we will substitute the obtained value of $x$ in either equation 1 or in equation 2. Now we will substitute value of $x$ i.e. 2 in equation 1 we will get,

$\begin{align}

& \Rightarrow 3=2\times (2)+y \\

& \Rightarrow y=3-4 \\

& \Rightarrow y=-1 \\

\end{align}$

We obtained the value of $x$ and $y$ by solving equation 1 and equation 2.

The values of $x$ and $y$ are 2 and -1 respectively.

$x=2,y=-1$.

Note: While solving a pair of linear equations in two variables first we will find the nature of a pair of linear equations in two variables and then, we proceed to solve if lines are not parallel.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Mark and label the given geoinformation on the outline class 11 social science CBSE

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Trending doubts

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference Between Plant Cell and Animal Cell

Which are the Top 10 Largest Countries of the World?

Write a letter to the principal requesting him to grant class 10 english CBSE

10 examples of evaporation in daily life with explanations

Give 10 examples for herbs , shrubs , climbers , creepers

Change the following sentences into negative and interrogative class 10 english CBSE