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Two sidewalks in a park are represented by lines on a coordinate grid. Two points on each of the lines are shown in the tables.
Sidewalk 1
x y
2 8
0 3
Sidewalk 2
x y
1 4
3 3
(a) Write the equation for Sidewalk 1 in slope-intercept form.
(b) Write the equation for Sidewalk 2 in point-slope form and then in slope-intercept form.
(c) Is the system of equations consistent independent, coincident, or inconsistent? Explain.
(d) If the two sidewalks intersect, what are the coordinates of the point of intersection? Use the substitution method and show your work.

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sqdancefan sqdancefan · Answer 1 · 2019-12-11T03:14:58+01:00

Answer:

(a) y = 5/2x +3

(b) y = -1/2x +9/2

(c) consistent, independent

(d) (0.5, 4.25)

Step-by-step explanation:

(a) The 2-point form of the equation of a line is a good place to start:

y = (y2 -y1)/(x2 -x1)(x -x1) +y1

Filling in the point values, we have ...

y = (8 -3)/(2 -0)(x -0) +3

y = 5/2x +3 . . . . . . simplify

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(b) Similarly, filling in the point values, we have ...

y = (3 -4)/(3 -1)(x -1) +4

y = -1/2(x -1) +4 . . . . . . simplify

y = -1/2x +9/2 . . . . . . . eliminate parentheses, collect terms

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(c) The lines have different slopes, so are consistent and independent.

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(d) Substituting for y, we have ...

5/2x + 3 = -1/2x + 9/2

5x +6 = -x +9 . . . . . . . . . . . . multiply by 2 to eliminate fractions

6x = 3 . . . . . . . . . . . . . . . . . . add x-6

x = 3/6 = 1/2 . . . . . . . . . . . . . divide by the x-coefficient

Using the equation for Sidewalk 2, we can find y:

y = -1/2(1/2) +9/2

y = 17/4

The point of intersection is (x, y) = (1/2, 17/4).