Question

One line passes through the points (-1, 6) and (2, 0). A second line passes through the points (-5, 5) and (- 1,9).Find the intersection point of the two lines.

Answer 1

Let's first find the equation of the first line given the points (-1,6 nd (2,0):

$\begin{gathered} \text{ Slope:} \\ m=\frac{y_2-y_1}{x_2-x_1}=\frac{0-6}{2-(-1)}=\frac{-6}{2+1}=\frac{-6}{3}=-2 \\ m=-2 \\ \text{Equation in point-slope form:} \\ y-y_2=m(x-x_2) \\ \Rightarrow y-0=-2(x-2)=-2x+4 \\ y=-2x+4 \end{gathered}$

For the second line, we have the points (-5,5) and (-1,9), then, the equation of the line is:

$\begin{gathered} \text{ Slope:} \\ m=\frac{9-5}{-1-(-5)}=\frac{4}{-1+5}=\frac{4}{4}=1 \\ m=1 \\ \text{ Equation in point-slope form:} \\ y-9=1(x-(-1))=x+1_{} \\ \Rightarrow y=x+1+9=x+10 \\ y=x+10 \end{gathered}$

Finally, we have the two equations:

$\begin{gathered} y=x+10 \\ y=-2x+4 \end{gathered}$

to find the point of intersection between them, we can equate both expressions to get the following:

$x+10=-2x+4$

solving for x we get:

$\begin{gathered} x+10=-2x+4 \\ \Rightarrow x+2x=4-10 \\ \Rightarrow3x=-6 \\ \Rightarrow x=-\frac{6}{3}=-2 \\ x=-2 \end{gathered}$

now that we have that x = -2, we can use this value in any of the equations to find the value of y:

$\begin{gathered} x=-2 \\ y=-2x+4 \\ \Rightarrow y=-2(-2)+4=4+4=8 \\ y=8 \end{gathered}$

therefore, the intersection point of the two lines is (-2,8)