Answer:
(3,1) is the correct answer.
Step-by-step explanation:
It is given that <em>first line </em>passes through (0,4) and (3,1)
Let the two points be at coordinates:
and ![(x_2,y_2)](https://tex.z-dn.net/?f=%28x_2%2Cy_2%29)
![x_1=0,\\y_1= 4,\\x_2=3,\\y_2= 1](https://tex.z-dn.net/?f=x_1%3D0%2C%5C%5Cy_1%3D%204%2C%5C%5Cx_2%3D3%2C%5C%5Cy_2%3D%201)
Equation of a line using two coordinates is given as:
![y = mx+c](https://tex.z-dn.net/?f=y%20%3D%20mx%2Bc)
![m = \dfrac{y_2-y_1}{x_2-x_1} = \dfrac{1-4}{3-0} = \dfrac{-3}{3} =-1](https://tex.z-dn.net/?f=m%20%3D%20%5Cdfrac%7By_2-y_1%7D%7Bx_2-x_1%7D%20%3D%20%5Cdfrac%7B1-4%7D%7B3-0%7D%20%3D%20%5Cdfrac%7B-3%7D%7B3%7D%20%3D-1)
Put x = 0, y = 4
![4 = 0 +c\\\Rightarrow c = 4](https://tex.z-dn.net/?f=4%20%3D%200%20%2Bc%5C%5C%5CRightarrow%20c%20%3D%204)
So, <em>first equation </em>is: ![y = -x +4 ...... (1)](https://tex.z-dn.net/?f=y%20%3D%20-x%20%2B4%20......%20%281%29)
It is given that <em>second line </em>passes through (0,-2) and (3,1)
Let the two points be at coordinates:
and ![(x_2,y_2)](https://tex.z-dn.net/?f=%28x_2%2Cy_2%29)
![x_1=0,\\y_1= -2,\\x_2=3,\\y_2= 1](https://tex.z-dn.net/?f=x_1%3D0%2C%5C%5Cy_1%3D%20-2%2C%5C%5Cx_2%3D3%2C%5C%5Cy_2%3D%201)
Equation of a line using two coordinates is given as:
![y = mx+c](https://tex.z-dn.net/?f=y%20%3D%20mx%2Bc)
![m = \dfrac{y_2-y_1}{x_2-x_1} = \dfrac{1-(-2)}{3-0} = \dfrac{3}{3} =1](https://tex.z-dn.net/?f=m%20%3D%20%5Cdfrac%7By_2-y_1%7D%7Bx_2-x_1%7D%20%3D%20%5Cdfrac%7B1-%28-2%29%7D%7B3-0%7D%20%3D%20%5Cdfrac%7B3%7D%7B3%7D%20%3D1)
Put x = 0, y = -2
![-2 = 0 +c\\\Rightarrow c = -2](https://tex.z-dn.net/?f=-2%20%3D%200%20%2Bc%5C%5C%5CRightarrow%20c%20%3D%20-2)
So, <em>second equation </em>is: ![y = x -2 ...... (2)](https://tex.z-dn.net/?f=y%20%3D%20x%20-2%20......%20%282%29)
<em>Solving equations</em> (1) and (2) using substitution:
Adding Equation (1) and (2):
![2y = 4-2\\\Rightarrow y =1](https://tex.z-dn.net/?f=2y%20%3D%204-2%5C%5C%5CRightarrow%20y%20%3D1)
Putting value y = 1 in equation (1):
![1 = -x+4\\\Rightarrow x =3](https://tex.z-dn.net/?f=1%20%3D%20-x%2B4%5C%5C%5CRightarrow%20%20x%20%3D3)
So, the solution for the system of equations is: <em>(3,1)</em>