The general equation of a stright line is,
[tex]y=mx+c[/tex]Here, m is the slope and c is the y intercept of the line. x, y are coordinates.
From the graph, the y intercept of the red line is c=3.
In the given equations, y=x+3 has y intercept as c=3.
So, the equation of the red line will be y=x+3.
The y intercept of the green line is c=-2.
So, the equation of the green line is y=-3x-2.
The y intercept of the violet line is c=1.
So, the equation of the violet line is y=2x+1.
Since the only remaining equation is y=-x-7, the equation of the brown line will be y=-x-7.
Now, from the graph, the point of intersection of y = 2x + 1 and y= x + 3 is (2, 5).
The point of intersection of y=x+3 and y=-3x-2 can be found by equating the expressions.
[tex]\begin{gathered} x+3=-3x-2 \\ x+3x=-2-3 \\ 4x=-5 \\ x=\frac{-5}{4} \end{gathered}[/tex]Now, put x=-5/4 in y=x+3 to find y.
[tex]\begin{gathered} y=x+3 \\ y=-\frac{5}{4}+3 \\ y=\frac{-5+3\times4}{4} \\ y=\frac{-5+12}{4} \\ y=\frac{7}{4} \end{gathered}[/tex]So, the point of intersection of y=x+3 and y=-3x-2 is (x,y)=(-5/4, 7/4).
From the graph, the point of intersection of y=-x-7 and x+3 is (-5, -2).
The point of intersection of y=-x-7and y=2x+1 can be found by equating the expressions.
[tex]\begin{gathered} -x-7=2x+1 \\ -x-2x=7+1 \\ -3x=8 \\ x=\frac{-8}{3} \end{gathered}[/tex]Now, put x=-8/3 in y=-x-7 to find y.
[tex]\begin{gathered} y=-(\frac{-8}{3})-7 \\ =\frac{8}{3}-7 \\ =\frac{8-7\times3}{3} \\ =\frac{8-21}{3} \\ =\frac{-13}{3} \end{gathered}[/tex]So, the point of intersection of y=-x-7 and y=2x+1 is (x,y)=(-8/3, -13/3).
The point of intersection of y=2x+1 and y=-3x-2 can be found by equating the expressions.
[tex]\begin{gathered} 2x+1=-3x-2 \\ 2x+3x=-2-1 \\ 5x=-3 \\ x=\frac{-3}{5} \end{gathered}[/tex]Now, put x=-3/5 in y=2x+1 to find y.
[tex]\begin{gathered} y=2\times(\frac{-3}{5})+1 \\ =\frac{-6}{5}+1 \\ =\frac{-6+5}{5} \\ =\frac{-1}{5} \end{gathered}[/tex]So, the point of intersection of y=2x+1 and y=-3x-2 is (-3/5, -1/5).
