Answer :
Let's tackle this problem step by step.
### (a) Write the equation for Sidewalk 1 in slope-intercept form
1. Identify the given points:
- The points given for Sidewalk 1 are (2, 7) and (0, 3).
2. Calculate the slope [tex]\( m \)[/tex]:
The formula for the slope between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substituting the values:
[tex]\[ m = \frac{3 - 7}{0 - 2} = \frac{-4}{-2} = 2 \][/tex]
3. Determine the y-intercept [tex]\( b \)[/tex]:
Use the slope-intercept form [tex]\( y = mx + b \)[/tex] and one of the points [tex]\((x_1, y_1)\)[/tex].
Substituting [tex]\( m = 2 \)[/tex], [tex]\( x_1 = 2 \)[/tex], and [tex]\( y_1 = 7 \)[/tex]:
[tex]\[ 7 = 2 \cdot 2 + b \implies 7 = 4 + b \implies b = 3 \][/tex]
4. Write the equation:
The slope-intercept form equation is:
[tex]\[ y = 2x + 3 \][/tex]
### (b) Write the equation for Sidewalk 2 in point-slope form and then in slope-intercept form
1. Identify the given points:
- The points given for Sidewalk 2 are (1, 5) and (3, 3).
2. Calculate the slope [tex]\( m \)[/tex]:
Using the slope formula:
[tex]\[ m = \frac{3 - 5}{3 - 1} = \frac{-2}{2} = -1 \][/tex]
3. Write the point-slope form equation:
The point-slope form of the equation is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Using point [tex]\((1, 5)\)[/tex] and slope [tex]\( m = -1 \)[/tex]:
[tex]\[ y - 5 = -1(x - 1) \implies y - 5 = -x + 1 \][/tex]
4. Convert to slope-intercept form:
Solving for [tex]\( y \)[/tex]:
[tex]\[ y = -x + 1 + 5 \implies y = -x + 6 \][/tex]
### (c) Is the system of equations consistent, independent, coincident, or inconsistent? Explain
1. Compare the slopes and y-intercepts of the two lines:
- Equation for Sidewalk 1: [tex]\( y = 2x + 3 \)[/tex]
- Equation for Sidewalk 2: [tex]\( y = -x + 6 \)[/tex]
2. Determine the relationship:
- The slopes are different ([tex]\(2\)[/tex] and [tex]\(-1\)[/tex]).
- Since the slopes are different, the lines are neither parallel nor coincident.
3. Conclusion:
- Since the slopes are different, the lines will intersect at exactly one point.
- Therefore, the system of equations is consistent and independent.
### (d) If the two sidewalks intersect, what are the coordinates of the point of intersection? Use the substitution method and show your work
1. Set the equations equal to each other:
[tex]\[ 2x + 3 = -x + 6 \][/tex]
2. Solve for [tex]\( x \)[/tex]:
[tex]\[ 2x + x = 6 - 3 \implies 3x = 3 \implies x = 1 \][/tex]
3. Substitute [tex]\( x = 1 \)[/tex] back into one of the equations to find [tex]\( y \)[/tex]:
Using the equation [tex]\( y = 2x + 3 \)[/tex]:
[tex]\[ y = 2(1) + 3 = 2 + 3 = 5 \][/tex]
4. Coordinate of the intersection point:
The two sidewalks intersect at the point [tex]\((1, 5)\)[/tex].
To summarize:
(a) The equation for Sidewalk 1 in slope-intercept form is [tex]\( y = 2x + 3 \)[/tex].
(b) The equation for Sidewalk 2 in point-slope form is [tex]\( y - 5 = -1(x - 1) \)[/tex], which converts to [tex]\( y = -x + 6 \)[/tex] in slope-intercept form.
(c) The system of equations is consistent and independent.
(d) The coordinates of the point of intersection are [tex]\((1, 5)\)[/tex].
### (a) Write the equation for Sidewalk 1 in slope-intercept form
1. Identify the given points:
- The points given for Sidewalk 1 are (2, 7) and (0, 3).
2. Calculate the slope [tex]\( m \)[/tex]:
The formula for the slope between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substituting the values:
[tex]\[ m = \frac{3 - 7}{0 - 2} = \frac{-4}{-2} = 2 \][/tex]
3. Determine the y-intercept [tex]\( b \)[/tex]:
Use the slope-intercept form [tex]\( y = mx + b \)[/tex] and one of the points [tex]\((x_1, y_1)\)[/tex].
Substituting [tex]\( m = 2 \)[/tex], [tex]\( x_1 = 2 \)[/tex], and [tex]\( y_1 = 7 \)[/tex]:
[tex]\[ 7 = 2 \cdot 2 + b \implies 7 = 4 + b \implies b = 3 \][/tex]
4. Write the equation:
The slope-intercept form equation is:
[tex]\[ y = 2x + 3 \][/tex]
### (b) Write the equation for Sidewalk 2 in point-slope form and then in slope-intercept form
1. Identify the given points:
- The points given for Sidewalk 2 are (1, 5) and (3, 3).
2. Calculate the slope [tex]\( m \)[/tex]:
Using the slope formula:
[tex]\[ m = \frac{3 - 5}{3 - 1} = \frac{-2}{2} = -1 \][/tex]
3. Write the point-slope form equation:
The point-slope form of the equation is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Using point [tex]\((1, 5)\)[/tex] and slope [tex]\( m = -1 \)[/tex]:
[tex]\[ y - 5 = -1(x - 1) \implies y - 5 = -x + 1 \][/tex]
4. Convert to slope-intercept form:
Solving for [tex]\( y \)[/tex]:
[tex]\[ y = -x + 1 + 5 \implies y = -x + 6 \][/tex]
### (c) Is the system of equations consistent, independent, coincident, or inconsistent? Explain
1. Compare the slopes and y-intercepts of the two lines:
- Equation for Sidewalk 1: [tex]\( y = 2x + 3 \)[/tex]
- Equation for Sidewalk 2: [tex]\( y = -x + 6 \)[/tex]
2. Determine the relationship:
- The slopes are different ([tex]\(2\)[/tex] and [tex]\(-1\)[/tex]).
- Since the slopes are different, the lines are neither parallel nor coincident.
3. Conclusion:
- Since the slopes are different, the lines will intersect at exactly one point.
- Therefore, the system of equations is consistent and independent.
### (d) If the two sidewalks intersect, what are the coordinates of the point of intersection? Use the substitution method and show your work
1. Set the equations equal to each other:
[tex]\[ 2x + 3 = -x + 6 \][/tex]
2. Solve for [tex]\( x \)[/tex]:
[tex]\[ 2x + x = 6 - 3 \implies 3x = 3 \implies x = 1 \][/tex]
3. Substitute [tex]\( x = 1 \)[/tex] back into one of the equations to find [tex]\( y \)[/tex]:
Using the equation [tex]\( y = 2x + 3 \)[/tex]:
[tex]\[ y = 2(1) + 3 = 2 + 3 = 5 \][/tex]
4. Coordinate of the intersection point:
The two sidewalks intersect at the point [tex]\((1, 5)\)[/tex].
To summarize:
(a) The equation for Sidewalk 1 in slope-intercept form is [tex]\( y = 2x + 3 \)[/tex].
(b) The equation for Sidewalk 2 in point-slope form is [tex]\( y - 5 = -1(x - 1) \)[/tex], which converts to [tex]\( y = -x + 6 \)[/tex] in slope-intercept form.
(c) The system of equations is consistent and independent.
(d) The coordinates of the point of intersection are [tex]\((1, 5)\)[/tex].
