Question

Show that the system of equations \[2x+5y=17,\text{ }5x+3y=14\] has a unique solution. Find the solution.

Answer 1

Hint: To have an unique solution between two line segment means that the lines intersect at a single coordinate and to find that coordinate we need to use the formula as:
 \[\begin{matrix}
   A\left( ax+by=c \right) \\
   a\left( Ax+By=C \right) \\
\end{matrix}\]
Subtract the common variable and then find the remaining variable and then after that place that variable to find the other variable.

Complete step-by-step answer:
Let us take the equations in form of \[ax+by=c\] and \[Ax+By=C\] i.e. for \[2x+5y=17\] :
The value of \[a=2\] , \[b=5\] and \[c=17\]
And for the equation \[5x+3y=14\] :
The value of \[A=5\] , \[B=3\] and \[C=14\]
After this we place the values in the formula as:
 \[\Rightarrow \begin{matrix}
   5\left( 2x+5y=17 \right) \\
   2\left( 5x+3y=14 \right)\text{ } \\
\end{matrix}\]
 \[\Rightarrow \begin{matrix}
   10x+25y=85 \\
   \begin{align}
  & 10x+6y=28\text{ } \\
 & -\text{ }-\text{ } \\
\end{align} \\
\end{matrix}\]
Subtracting the values of the above common variable we get the value of \[y\] component as:
 \[\Rightarrow 10x-10x+25y-6y=85-28\]
 \[\Rightarrow 19y=57\]
 \[\Rightarrow y=3\]
Placing the value of \[y\] in the equation \[2x+5y=17\] , we get the value of \[x\] as:
 \[\Rightarrow 2x+5\times 3=17\]
 \[\Rightarrow 2x=2\]
 \[\Rightarrow x=1\]
Hence, the coordinate at which we get the unique solution is at \[x=1\] and \[y=3\] .
So, the correct answer is “ \[x=1\] and \[y=3\] ”.

Note: Apart from unique solution there are two solutions which are No solution and Infinite solution, no solution meaning the line segment doesn’t intersect with each other at any point and infinite solution meaning all points from the two lines are intersecting each other at every point.