Question

A line passes through the points (1, -1) and (5, 3). A second line passes through the points (4, 7) and (8, 1). At what point do the two lines intersect? O A. (0, 13) O B. (5,8) 0 C. (3,6) D. (6,4)

Answer 1

First, find the equation of each line. Next, solve the system of equations to find the point at which those lines intersect.

The general equation of a line in slope-intercept form, is:

`y=mx+b`

Where m is the slope of the line and b is the y-intercept.

On the other hand, given the coordinates of two points:

`\begin{gathered} (x_1,y_1) \\ (x_2,y_2) \end{gathered}`

The slope of a line through those two points is given by:

`\begin{gathered} m=(\Delta y)/(\Delta x) \\ \Rightarrow m=(y_2-y_1)/(x_2-x_1) \end{gathered}`

Find the equation of the first line in slope-intercept form. Using the coordinates (1,-1) and (5,3), calculate the slope:

`\begin{gathered} m_1=(3-(-1))/(5-1) \\ =(3+1)/(4) \\ =(4)/(4) \\ =1 \end{gathered}`

Substitute m=1 into the equation of a line in slope-intercept form:

`\begin{gathered} y=1\cdot x+b \\ \Rightarrow y=x+b \end{gathered}`

Substitute the coordinates of a point into the equation to find the value of b. Use the point (1,-1), so that x=1 and y=-1:

`\begin{gathered} y=x+b \\ \Rightarrow-1=1+b \\ \Rightarrow b=-2 \end{gathered}`

Therefore, the equation of the first line, is:

`y=x-2`

Using a similar method, we can find that the slope of the second line is:

`\begin{gathered} m=(7-1)/(4-8) \\ =(6)/(-4) \\ =-(3)/(2) \end{gathered}`

And the y-intercept will be given by:

`\begin{gathered} 7=-(3)/(2)(4)+b \\ \Rightarrow b=13 \end{gathered}`

Therefore, the equation of the second line, is:

`y=-(3)/(2)x+13`

Substitute y=x-2 from the first equation into y in the second equation and solve for x to find the x-coordinate of the point at which these lines intersect.

`\begin{gathered} y=x-2 \\ y=-(3)/(2)x+13 \\ \Rightarrow x-2=-(3)/(2)x+13 \\ \Rightarrow2x-4=-3x+26 \\ \Rightarrow5x-4=26 \\ \Rightarrow5x=30 \\ \Rightarrow x=6 \end{gathered}`

Substitute x=6 into y=x-2 to find the y-coordinate of the point at which these lines intersect:

`\begin{gathered} y=6-2 \\ \Rightarrow y=4 \end{gathered}`

Since x=6 and y=4, therefore, the point at which these lines intersect, is:

`(6,4)`