Need an answer ASAP

Question

the points (3,1) and (1,-3) form a line (line a). Identify which line below is parallel to this line and which line is perpendicular to this line. At what point do the perpendicular lines intersect?

Line b: -4x+2y=14

Line c: x+2y=0

Answer 1

Every line in a two coordinate system (x-axis & y-axis) will have a slope. Why? Because one coordinate is a function of the other: y = function of x.

That said, lets describes all three lines as linear functions using the equation y = m * x + b where m is the ratio or slope between y and x values and b is the y-intercept ( 0 , b )

Line a is described as two point: ( 3 , 1 ) & ( 1 , -3 ). We can describe the slope, m in the following way:

m = ( y₁ - y₂ ) / ( x₁ - x₂ )

We will call (3,1) = (x₁,y₁) & (1,-3)=(x₂,y₂)

m_a = ( 1 - (-3) ) / ( 3 - 1 ) = ( 1 + 3 ) / 2 = 4 / 2 = 2

y = 2 * x + b

To find b simply plug in one of the points into the equation (either one is fine since both are on the same slope)

1 = 2 * 3 + b

1 = 6 + b

b = - 5

Our final equation for line a is:

y = 2 * x - 5

Line b is described as -4 * x + 2 * y = 14. We can reformat this equation as y = m * x + b by using our arithmetic techniques:

-4 * x + 2 * y = 14

2 * y = 4 * x + 14

y = 2 * x + 7

2 must equal m so for line b: m_b = 2

Line c is described as x + 2 y = 0. As with part b we can reformat this equation to get:

y = -1/2 * x

so m_c = - 1 / 2

We can conclude the following:

m_b is the same sign and value of m_a so it must be parallel to the line (aka slope) of m_a

m_c is the opposite sign and the reciprocal of m_a so it must be perpendicular to the line (aka slope) of m_a

Finally at what point do the perpendicular lines intersect? Simply let the function a equal to function c:

-1/2 * x_intersect = 2 * x_intersect - 5

solve for x:

x = -4 * x + 10

5 * x = 10

x_intersect = 2

Now plug in x = 2 into either function of a or function of c to find y

y = -1

The point of intersection is (2, -1)

Answer 2

The slope of the line through the two given points is [1 - (-3)]/(3 -1) = 2.

The slope of line b is also 2; therefore, b is parallel

The slope of line c is -1/2 and, therefore, perpendicular to a.

Now get the equation of line a using the slope of 2.

You then need to solve the pair of equations for a and c and then for b and c. If you are unable to do that, please let me know.