Question

Use the given points of line a and line b to determine whether the lines intersect, are parallel, or coincide.Line a: (1,5) and (4,14) line b: (-2,5) and (6,-27)

Answer 1

To solve this problem, the first step is to find the equation for both line a and line b.

Use the given points to find the slope.

$\begin{gathered} m=\frac{y2-y1}{x2-x1} \\ m_a=\frac{14-5}{4-1}=\frac{9}{3}=3 \\ m_b=\frac{-27-5}{6-(-2)}=-\frac{32}{8}=-4 \end{gathered}$

Use the slope and one of the given points to find the equation of the lines. Use point slope formula and solve for y to find the slope intercept form equations.

$\begin{gathered} y-y1=m(x-x1) \\ \\ y-5=3\mleft(x-1\mright) \\ y=3x-3+5 \\ y=3x+2 \\ \\ y-5=-4(x-(-2)) \\ y=-4x-8+5 \\ y=-4x-3 \end{gathered}$

For 2 lines to be parallel, they need to have the same slope. In this case, lines a and b don't have the same slope, which means they are not parallel.

For 2 lines to coincide, they need to have exactly the same equation. In this case, they don't have the same equation, which means they do not coincide.

The last option is to check if the lines intersect. To do this, find the solution of the system of equations formed by the equations of both lines. Use the equalization method.

$\begin{gathered} 3x+2=-4x-3 \\ 3x+4x=-3-2 \\ 7x=-5 \\ x=-\frac{5}{7} \\ y=3(-\frac{5}{7})+2 \\ y=-\frac{15}{7}+2 \\ y=-\frac{1}{7} \end{gathered}$

It means, lines a and b intersect at (-5/7,-1/7).