(Score for Question 3: ___ of 5 points) 3. 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 7 0 3 Sidewalk 2 x y 1 5 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.

Question

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Answer 1

(a) For Sidewalk 1, using points (2, 7) and (0, 3):

1. Find the slope (m):
   \( m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 7}{0 - 2} = \frac{-4}{-2} = 2 \)

2. Use point-slope form:
   \( y - 7 = 2(x - 2) \)
   Expand:
   \( y = 2x + 3 \)

   Equation: \( y = 2x + 3 \)

(b) For Sidewalk 2, using points (1, 5) and (3, 3):

1. Find the slope (m):
   \( m = \frac{3 - 5}{3 - 1} = \frac{-2}{2} = -1 \)

2. Use point-slope form:
   \( y - 5 = -1(x - 1) \)
   Expand:
   \( y = -x + 6 \)

   Equation in slope-intercept form: \( y = -x + 6 \)

(c) The system of equations is consistent and independent because the slopes are different (2 and -1), meaning they intersect at one point.

(d) To find the point of intersection, set the equations equal:

\( 2x + 3 = -x + 6 \)
\( 2x + x = 6 - 3 \)
\( 3x = 3 \)
\( x = 1 \)

Now substitute \( x = 1 \) back into either equation to get \( y \):

Using \( y = 2x + 3 \):
\( y = 2(1) + 3 = 5 \)

Coordinates of intersection: \( (1, 5) \)