Question

Solve the following systems of linear equations. Also find the coordinates of the points where the lines meet the axis of y.
$3x + y - 5 = 0$,
$2x - y - 5 = 0$.
$
  A.{\text{ }}x = 2,y = - 1;\left( {0,5} \right),\left( {1, - 5} \right) \\
  B.{\text{ }}x = 2,y = 1;\left( {1,5} \right),\left( {0, - 5} \right) \\
  C.{\text{ }}x = 2,y = 1;\left( {0,5} \right),\left( {1, - 5} \right) \\
  D.{\text{ }}x = 2,y = - 1;\left( {0,5} \right),\left( {0, - 5} \right) \\
$

Answer 1

Hint – In this particular question from any equation write any variable in terms of another variable and substitute this value in another equation later on substitute x = 0 to calculate the coordinates on y-axis so use this concept to reach the solution of the question.

Complete step-by-step answer:
Line – 1; $3x + y - 5 = 0$
Line – 2; $2x - y - 5 = 0$
We use the substitution method to solve this system of equations.
So, from line 1, $y = 5 - 3x$ ................. (1)
So substitute this in line 2 we have,
$ \Rightarrow \left( {2x - \left( {5 - 3x} \right) - 5 = 0} \right)$
Now simplify this equation we have,
$ \Rightarrow 2x - 5 + 3x - 5 = 0$
$ \Rightarrow 5x = 10$
$ \Rightarrow x = 2$
Now from equation (1)
\[ \Rightarrow y = 5 - 3\left( 2 \right) = 5 - 6 = - 1\]
So the solution is (x, y) = (2, -1).
Now the intersection point of the lines with the y-axis are
Line $3x + y - 5 = 0$, as we know that on y-axis x-coordinate is zero so put x = 0 in the line we have,
\[ \Rightarrow \left( {3\left( 0 \right) + y - 5 = 0} \right)\]
$ \Rightarrow y - 5 = 0$
$ \Rightarrow y = 5$
So the intersection point with the y-axis is (0, 5).
Now another line $2x - y - 5 = 0$, as we know that on y-axis x-coordinate is zero so put x = 0 in the line we have,
\[ \Rightarrow \left( {2\left( 0 \right) - y - 5 = 0} \right)\]
$ \Rightarrow - y - 5 = 0$
$ \Rightarrow y = - 5$
So the intersection point with the y-axis is (0, -5).
So this is the required answer.
Hence option (D) is the correct answer.

Note – Whenever we face such types of questions we can solve these equations using any method (i.e. by graphically, cross multiplication method, substitution method, elimination method etc.) so here we use substitution method which is shown above and the coordinates on y-axis is found out by substituting x = 0 (because on y-axis the coordinate of x is always zero) as above in the given system of linear equation so just simplify we will get the required answer.