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Anonymous asked in Science & MathematicsMathematics · 2 months ago

# find the value of r so the line passes that through (-5,2) and (3,r) has a slope of -1/2?

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• The typical equation of a line is: y = mx + y₀ → where m: slope and where y₀: y-intercept

y = mx + y₀ ← this is the equation of the line (ℓ)

The line has a slope of (- 1/2), so you can say that: m = - 1/2

y = - (1/2).x + y₀

The line passes through (- 5 ; 2), so these coordinates must verify the equation of the line.

y = - (1/2).x + y₀

y₀ = y + (1/2).x → you substitute x and y by the coordinates of the point (- 5 ; 2)

y₀ = 2 + [(1/2) * - 5]

y₀ = 2 - (5/2)

y₀ = - 1/2

Recall the equation of the line:

y = - (1/2).x + y₀ → we've just seen that: y₀ = - 1/2

y = - (1/2).x - (1/2) → to go further

y = (- x - 1)/2

2y = - x - 1

x + 2y = - 1 ← this is the equation of the line (ℓ)

x + 2y = - 1 → as the point (3 ; r) belongs to the line, its coordinates verify the equation

3 + 2r = - 1

2r = - 4

r = - 2

• (r-2)/(3-(-5))=-1/2

=>

r-2=8(-1/2)

=>

r-2=-4

=>

r=-2

• Let's apply the slope formula to solve for r..

.........y₂ - y₁

m = --------------

.........x₂ - x₁₂

............r - 2

-1/2 = ---------

............3 - (-5)

..............r - 2

- 1/2 = ------------

................8

- 4 = r - 2

- 4 + 2 = r

•  The value of r so that the line passes through (-5, 2) and (3, r)

and has a slope of -1/2:

(-5, 2), (3, r)

m = -1/2

(2 -  r) / (-5 - 3) = -1/2

2(2 - r) = -1(-5 - 3)

4 - 2r = 8

2r = -4

r = -2

• -1/2 = (r - 2)/(3 - (-5)) ===> -1/2 = (r - 2)/(3 + 5) ===> -1/2 = (r - 2)/8 ===>

===> -4 = r - 2 ===> -4 + 2 = r ===> r = -2 (ANSWER)

• The value of r = -2 because the slope is (2--2)/(-5-3) = -1/2 and the points are (-5, 2) and (3, -2)

• The slope is a change in y over the change in x.  So we can set up this equation:

m = (y₁ - y₂) / (x₁ - x₂)

Using the two points: (-5, 2) and (3, r) for the x's and y's and m = -1/2, we can substitute this into the equation and solve for the unknown:

-1/2 = (2 - r) / (-5 - 3)

-1/2 = (2 - r) / (-8)

4 = 2 - r

2 = -r

-2 = r