Question

How to determine the number of solutions in a linear system without solving: $14x+3y=12$ and $14x-3y=0$?

Answer 1

Hint: In this question given we have been asked to find the number of solutions of the given system of linear equations are $14x+3y=12$ and $14x-3y=0$.We can find the number of solutions for the above given system of linear equations by finding the slopes of the given equations. We know that if a straight line equation is in the form of $ax+by=c$ then its slope can be determined by the formula $-\dfrac{a}{b}$

Complete step by step answer:
Now considering from the question we have a linear system consisting of $14x+3y=12$ and $14x-3y=0$.
We know that if a straight line equation is in the form of $ax+by=c$ then its slope can be determined by the formula $-\dfrac{a}{b}$
Now, by using the above formula of the slope we can find the slopes of the above system of linear equations.
First of all we find the slope of the equation $14x+3y=12$
Given equation is $14x+3y=12$
We can clearly observe that it is in the form of $ax+by=c$
By comparing the coefficients we get,
$\begin{align}
  & a=14 \\
 & b=3 \\
\end{align}$
As we have been already discussed earlier slope of the straight line is $-\dfrac{a}{b}$
The slope will be given as $-\dfrac{14}{3}$
Now we find the slope of the second equation $14x-3y=0$
We can clearly observe that it is in the form of $ax+by=c$
By comparing the coefficients we get,
$\begin{align}
  & a=14 \\
 & b=-3 \\
\end{align}$
Slope of the equation is $\dfrac{14}{3}$
We can clearly observe that slopes of the both equations are different. There is a chance for two lines to cut each other.
As the slopes of the two lines are not equal, the system of equations has only one solution which will be the point of intersection of the lines which can be obtained by placing $14x=3y$ in $14+3y=12$ which was obtained from $14x-3y=0$.
After that we will have
$\begin{align}
  & 3y+3y=12 \\
 & \Rightarrow 6y=12 \\
 & \Rightarrow y=2 \\
\end{align}$
And by using the value of $x$ in any of the given equations we will have
$\begin{align}
  & 14x=3\left( 2 \right) \\
 & \Rightarrow x=\dfrac{6}{14} \\
 & \Rightarrow x=\dfrac{3}{7} \\
\end{align}$

Therefore we can conclude that there is a one solution that is $\left( \dfrac{3}{7},2 \right)$.

Note: We should be well aware of the straight lines and their general form of the equations. We should be very careful while calculating the slope of the system of the linear equations. We should be very careful while doing the calculation part. We should be well aware of the all formula of the straight lines so that they are very helpful while solving the problems. Here while answering this question we may not need to use the slope concept and find the solution by finding the intersection of the 2 lines.