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

Lines a and b have different slopes of 3 and -4, respectively. Therefore, they are neither parallel nor coinciding; they will intersect at one point.

Step-by-step explanation:

To determine whether the lines given by the sets of points intersect, are parallel, or coincide, we must first calculate the slope of each line. Then we can compare the slopes to decide the relationship between the lines.

For line a, passing through points (1,5) and (4,14), the slope m is calculated by the formula m = (y2 - y1) / (x2 - x1). So, m = (14 - 5) / (4 - 1) = 9 / 3 = 3.

For line b, passing through points (-2,5) and (6,-27), the slope m is calculated similarly: m = (-27 - 5) / (6 - (-2)) = -32 / 8 = -4.

Since the slopes of line a and line b are not equal, they are not parallel or coinciding lines. With different slopes, we can conclude that they will intersect at a single point.

Answer 2

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=(y2-y1)/(x2-x1) \\ m_a=(14-5)/(4-1)=(9)/(3)=3 \\ m_b=(-27-5)/(6-(-2))=-(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=-(5)/(7) \\ y=3(-(5)/(7))+2 \\ y=-(15)/(7)+2 \\ y=-(1)/(7) \end{gathered}`

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