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:

$\begin{gathered} m=\frac{\Delta y}{\Delta x} \\ \Rightarrow m=\frac{y_2-y_1}{x_2-x_1} \end{gathered}$

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

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

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:

$\begin{gathered} m=\frac{7-1}{4-8} \\ =\frac{6}{-4} \\ =-\frac{3}{2} \end{gathered}$

And the y-intercept will be given by:

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

Therefore, the equation of the second line, is:

$y=-\frac{3}{2}x+13$

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.

$\begin{gathered} y=x-2 \\ y=-\frac{3}{2}x+13 \\ \Rightarrow x-2=-\frac{3}{2}x+13 \\ \Rightarrow2x-4=-3x+26 \\ \Rightarrow5x-4=26 \\ \Rightarrow5x=30 \\ \Rightarrow x=6 \end{gathered}$

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:

$(6,4)$