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Determine the slope of lines a and h

by Lisa

Parallel, Perpendicular, or Intersecting Lines?

Given line “A” with points A (4,4) and B(8,10) and line “H” with points D(-2,3)
and E (0,6),

Determine the slope of line “A” and “H”.

Write the equation of line “A” and “H”.

Are the two lines parallel, perpendicular, or intersecting, at a non-right angle?
Explain your reasoning.

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Dec 23, 2012

Parallel, Perpendicular, or Intersecting Lines
by: Staff

Answer

Part I

Determine the slope of line “A”.

slope = "rise" over "run", or "rise" DIVIDED BY "run"

slope = (change in "y" values)/(change in "x" values)

slope = Δy / Δx

slope = (y₂ - y₁)/(x₂ - x₁)

The x-y coordinates for the two points for Line “A” are (4, 4) and (8, 10).

These points are already listed in the proper order (left to right)

They are listed properly as (4, 4) and (8, 10) because the slope will be calculated as the change of "y" when x increases. In this case "x" will increase from 4 to 8 (left to right).

It is worth plotting the two points so the position of each can be visualized.

$Linear Equation “A” – Slope of line Passing Through Two Points$

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Dec 23, 2012

Parallel, Perpendicular, or Intersecting Lines
by: Staff

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Part II

Calculation of the slope for Line “A”

Points: Left and Right

(4, 4) and (8, 10)

(x₁, y₁) and (x₂,y₂)

x₁ = 4
x₂ = 8

y₁ = 4
y₂ = 10

slope = (y₂ - y₁)/(x₂ - x₁)

slope = (10 - 4)/(8 - 4)

slope = (6)/(4)

slope = 6/4

slope = 3/2

Line “A” slope = 3/2

Write the equation of line “A”.

The Point-Slope Form

Point-Slope Form: (y - y₁) = m (x - x₁)

This format uses a single known point (x₁,y₁) and the slope m (which is also a known value)

From the problem statement:

m (the slope) = 3/2

single known point (x₁,y₁) = (4,4)

x₁ = 4

y₁ = 4

(y - y₁) = m (x - x₁)

(y - 4) = (3/2)*(x - 4)

Point Slope Form of Equation for Line “A”: (y - 4) = (3/2)*(x - 4)

The Slope Intercept Form

Slope Intercept Form

y = mx + b

(y - 4) = (3/2)*(x - 4)

2*(y - 4) = 2*(3/2)*(x - 4)

2y - 8 = 3x - 8

2y - 8 + 8 = 3x - 12 + 8

2y + 0 = 3x - 4

2y = 3x - 4

2y / 2 = (3x - 4) / 2

y * (2 / 2) = (3x - 4) / 2

y * (1) = (3x - 4) / 2

y = (3x - 4) / 2

y = (3/2)x - 2

Slope Form of Equation for Line “A”: y = (3/2)x - 2

The Standard Form

Standard Form

Ax + By = C

A, B, and C are constants

y = (3/2)x - 2

(3/2)x - 2 = y

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

(3/2)x + 0 = y + 2

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

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

(3/2)x - y = 0 + 2

(3/2)x - y = 2

Standard Form of Equation for Line “A”: (3/2)x - y = 2

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Dec 23, 2012

Parallel, Perpendicular, or Intersecting Lines
by: Staff

--------------------------------------------

Part III

Determine the slope of line “H”.

slope = "rise" over "run", or "rise" DIVIDED BY "run"

slope = (change in "y" values)/(change in "x" values)

slope = Δy / Δx

slope = (y₂ - y₁)/(x₂ - x₁)

The x-y coordinates for the two points for Line “H” are (-2, 3) and (0, 6).

These points are also listed in the proper order (left to right)

They are listed properly as (-2, 3) and (0, 6). The slope for Line “H” will be calculated in exactly the same way as the slope for Line “A” was calculated: the change of "y" when x increases. In this case "x" will increase from -2 to 3 (left to right).

The two points are plotted below.

$Linear Equation “H” – Slope of line Passing Through Two Points$

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Dec 23, 2012

Parallel, Perpendicular, or Intersecting Lines
by: Staff

--------------------------------------------

Part IV

Calculation of the slope for Line “H”

Points: Left and Right

(-2, 3) and (0, 6)

(x₁, y₁) and (x₂,y₂)

x₁ = -2
x₂ = 0

y₁ = 3
y₂ = 6

slope = (y₂ - y₁)/(x₂ - x₁)

slope = (6 - 3)/(0 - (-2))

slope = (3)/(2)

slope = 3/2

Line “H” slope = 3/2

Write the equation of line “H”.

The Point-Slope Form

Point-Slope Form: (y - y₁) = m (x - x₁)

This format uses a single known point (x₁,y₁) and the slope m (which is also a known value)

From the problem statement:

m (the slope) = 3/2

single known point (x₁,y₁) = (-2,3)

x₁ = -2

y₁ = 3

(y - y₁) = m (x - x₁)

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

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

Point Slope Form of Equation for Line “H”: (y - 3) = (3/2)*(x + 2)

The Slope Intercept Form

Slope Intercept Form

y = mx + b

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

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

2y - 6 = 3x + 6

2y - 6 + 6 = 3x + 6 + 6

2y + 0 = 3x + 12

2y = 3x + 12

2y / 2 = (3x + 12) / 2

y * (2 / 2) = (3x + 12) / 2

y * (1) = (3x + 12) / 2

y = (3x + 12) / 2

y = (3/2)x + 6

Point Slope Form of Equation for Line “H”: y = (3/2)x + 6

The Standard Form

Standard Form

Ax + By = C

A, B, and C are constants

y = (3/2)x + 6

(3/2)x + 6 = y

(3/2)x + 6 - 6 = y - 6

(3/2)x + 0 = y - 6

(3/2)x = y - 6

(3/2)x - y = y - 6 - y

(3/2)x - y = y - 6 - y

(3/2)x - y = y - y - 6

(3/2)x - y = 0 - 6

(3/2)x - y = -6

Standard Form of Equation for Line “H”: (3/2)x - y = -6

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Dec 23, 2012

Parallel, Perpendicular, or Intersecting Lines
by: Staff

--------------------------------------------

Part V

Are the two lines parallel, perpendicular, or intersecting, at a non-right angle? Explain your reasoning.

Line “A” and Line “H” are: parallel, since they have the same slope

$Math – Parallel Linear Equations$

Thanks for writing.

Staff
www.solving-math-problems.com

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Determine the slope of lines a and h

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