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Line Equation Examples

What is the distance between the crossing point of two lines and the origin?

Question 1

The crossing point between the two lines, 2x - y - 1 = 0 and x + 2y - 3 = 0 is d distance away from the origin. What is the value of d?

Solution

Find the crossing point by treating these two lines with regard to their corresponding equations. Multiplying the first equation by 2 and adding it to the second will give you 5x = 5, or x = 1. Then, plugging this value into either equation will give y = 1, so the point of crossing is (1, 1) and its distance to the origin is square root of (1 + 1) = square root of 2 = 1.414

2x - y - 1 = 0 ...equation1

x + 2y - 3 = 0 ...equation2

equation1 × 2: 4x - 2y - 2 = 0 ...equation3

equation3 + equation2: 4x + x - 2y + 2y - 2 + (-3) = 0

5x - 5 = 0

5x = 5

x = 1

from equation1: y = 2x - 1 = 2 × 1 - 1 = 1

the solutions are: x = 1 and y = 1

The distance from (x, y) = (1, 1) to the origin is:

Therefore, the distance to the origin is 1.414 units.

What is the Midpoint of a segment?

Question 2

Where are the coordinates of the midpoint M of the segment joing the pair of points (3, 5) and (-1, -7)?

Solution

The midpoint M(x, y) of the line segment is given by x = (x₁ + x₂)/2 and y = (y₁ + y₂)/2

So, the coordinates of the midpoint are

x = (x₁ + x₂)/2 = [3 + (-1)]/2 = (3 - 1)/2 = 2/2 = 1

y = (y₁ + y₂)/2 = [5 + (-7)]/2 = (5 - 7)/2 = -2/2 = -1

Therefore, the coordinate of the midpoint is (1, -1)

What is the line equation when two lines parallel?

Question 3

Obtain the equation of the line L₁ which passing through point ( -3, -1 ) and is parallel to L₂ , whose equatoion is 2x - y - 3 = 0.

Solution

Step 1:

Obtain the slope k₂ of line L₂ by solving the equation 2x - y - 3 = 0 for y in terms of x.

y = 2x - 3.

The slope-intercept form for one line is:

y = kx + b, then the slope of L₂ is k₂ = 2.

Step 2:

Two different nonvertical lines with slope k₁ and k₂ are parallel if and only if k₁ = k₂ .

Thus the slope of line L₁ is k₁ = 2.

Step 3:

To obtain line equation L₁ in slope-intercept form, using the formula:

y - y₀ = k(x - x₀).

Given x₀ = -3 and y₀ = -1,

then y - (-1) = 2[x - (-3)]

y + 1 = 2(x + 3)

y + 1 = 2x + 6

y = 2x + 5.

The general form of line L₁ is: 2x - y + 5 = 0.

What is the line equation when two lines are perpendicular?

Question 4

Line L₁ contains the point ( -3, 2 ) and is perpendicular to the L₂, whose equatoion is 3x + 2y - 6 = 0.

Solution

Step 1:

Obtain the slope k₂ of L₂ by solving the equation 3x + 2y - 6 = 0 for y in terms of x.

2y = -3x + 6

y = -(3/2)x + 3

The slope-intercept form for one line is:

y = kx + b, then the slope of L₂ is k₂ = -3/2.

Step 2:

Two nonvertical lines with slope k₁ and k₂ are perpendicular if and only if k₁ k₂ = -1,

or, equivalently, k₁ = - 1/k₂.

Thus the slope of L₁ is k₁ = -1/(-3/2) = 2/3.

Step 3:

To obtain line equation L₁ in slope-intercept form, using the formula:

y - y₀ = k(x - x₀).

Given x₀ = -3 and y₀ = 2,

then y - 2 = 2/3[x - (-3)]

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

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

y = (2/3)x + 4

both sides multiply 3 to remove denominator

3y = 2x + 12.

The general form of line L₁ is: 2x - 3y + 12 = 0.