Answer :
Certainly! Let's go through each part step by step.
### (a) Write the equation for Sidewalk 1 in slope-intercept form.
We need to find the equation of the line passing through the points (2, 7) and (0, 3). The slope-intercept form of a line is given by [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
1. Calculate the slope [tex]\( m \)[/tex]:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{7 - 3}{2 - 0} = \frac{4}{2} = 2 \][/tex]
2. The y-intercept [tex]\( b \)[/tex] is the value of [tex]\( y \)[/tex] when [tex]\( x = 0 \)[/tex], which is given directly as 3.
3. Thus, the equation for Sidewalk 1 is:
[tex]\[ y = 2x + 3 \][/tex]
Answer for (a): [tex]\( y = 2x + 3 \)[/tex]
### (b) Write the equation for Sidewalk 2 in point-slope form and then in slope-intercept form.
We need to find the equation of the line passing through the points (1, 5) and (3, 3).
1. Calculate the slope [tex]\( m \)[/tex]:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 5}{3 - 1} = \frac{-2}{2} = -1 \][/tex]
2. Using the point-slope form formula [tex]\( y - y_1 = m(x - x_1) \)[/tex] with point (1, 5):
[tex]\[ y - 5 = -1(x - 1) \][/tex]
Point-slope form equation: [tex]\( y - 5 = -1(x - 1) \)[/tex]
3. Convert to slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[ y - 5 = -x + 1 \][/tex]
[tex]\[ y = -x + 6 \][/tex]
Slope-intercept form equation: [tex]\( y = -x + 6 \)[/tex]
Answer for (b): Point-slope form: [tex]\( y - 5 = -1(x - 1) \)[/tex], Slope-intercept form: [tex]\( y = -x + 6 \)[/tex]
### (c) Is the system of equations consistent independent, coincident, or inconsistent? Explain.
To determine the consistency of the system, we compare the slopes and y-intercepts of the two equations:
- The equation for Sidewalk 1: [tex]\( y = 2x + 3 \)[/tex]
- The equation for Sidewalk 2: [tex]\( y = -x + 6 \)[/tex]
The slopes are different ([tex]\(m_1 = 2\)[/tex] and [tex]\(m_2 = -1\)[/tex]), so the lines are neither parallel nor coincident. Therefore, the system of equations is consistent and independent, meaning the lines intersect at exactly one point.
Answer for (c): The system of equations is consistent independent.
### (d) If the two sidewalks intersect, what are the coordinates of the point of intersection? Use the substitution method and show your work.
To find the intersection point, we solve the equations simultaneously using substitution:
1. Equation from Sidewalk 1: [tex]\( y = 2x + 3 \)[/tex]
2. Equation from Sidewalk 2: [tex]\( y = -x + 6 \)[/tex]
Substitute [tex]\( y \)[/tex] from the first equation into the second equation:
[tex]\[ 2x + 3 = -x + 6 \][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ 2x + 3 = -x + 6 \][/tex]
[tex]\[ 2x + x = 6 - 3 \][/tex]
[tex]\[ 3x = 3 \][/tex]
[tex]\[ x = 1 \][/tex]
Now, substitute [tex]\( x = 1 \)[/tex] back into the first equation to find [tex]\( y \)[/tex]:
[tex]\[ y = 2(1) + 3 = 2 + 3 = 5 \][/tex]
Hence, the coordinates of the point of intersection are [tex]\( (1, 5) \)[/tex].
Answer for (d): The coordinates of the point of intersection are [tex]\( (1, 5) \)[/tex].
### (a) Write the equation for Sidewalk 1 in slope-intercept form.
We need to find the equation of the line passing through the points (2, 7) and (0, 3). The slope-intercept form of a line is given by [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
1. Calculate the slope [tex]\( m \)[/tex]:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{7 - 3}{2 - 0} = \frac{4}{2} = 2 \][/tex]
2. The y-intercept [tex]\( b \)[/tex] is the value of [tex]\( y \)[/tex] when [tex]\( x = 0 \)[/tex], which is given directly as 3.
3. Thus, the equation for Sidewalk 1 is:
[tex]\[ y = 2x + 3 \][/tex]
Answer for (a): [tex]\( y = 2x + 3 \)[/tex]
### (b) Write the equation for Sidewalk 2 in point-slope form and then in slope-intercept form.
We need to find the equation of the line passing through the points (1, 5) and (3, 3).
1. Calculate the slope [tex]\( m \)[/tex]:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 5}{3 - 1} = \frac{-2}{2} = -1 \][/tex]
2. Using the point-slope form formula [tex]\( y - y_1 = m(x - x_1) \)[/tex] with point (1, 5):
[tex]\[ y - 5 = -1(x - 1) \][/tex]
Point-slope form equation: [tex]\( y - 5 = -1(x - 1) \)[/tex]
3. Convert to slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[ y - 5 = -x + 1 \][/tex]
[tex]\[ y = -x + 6 \][/tex]
Slope-intercept form equation: [tex]\( y = -x + 6 \)[/tex]
Answer for (b): Point-slope form: [tex]\( y - 5 = -1(x - 1) \)[/tex], Slope-intercept form: [tex]\( y = -x + 6 \)[/tex]
### (c) Is the system of equations consistent independent, coincident, or inconsistent? Explain.
To determine the consistency of the system, we compare the slopes and y-intercepts of the two equations:
- The equation for Sidewalk 1: [tex]\( y = 2x + 3 \)[/tex]
- The equation for Sidewalk 2: [tex]\( y = -x + 6 \)[/tex]
The slopes are different ([tex]\(m_1 = 2\)[/tex] and [tex]\(m_2 = -1\)[/tex]), so the lines are neither parallel nor coincident. Therefore, the system of equations is consistent and independent, meaning the lines intersect at exactly one point.
Answer for (c): The system of equations is consistent independent.
### (d) If the two sidewalks intersect, what are the coordinates of the point of intersection? Use the substitution method and show your work.
To find the intersection point, we solve the equations simultaneously using substitution:
1. Equation from Sidewalk 1: [tex]\( y = 2x + 3 \)[/tex]
2. Equation from Sidewalk 2: [tex]\( y = -x + 6 \)[/tex]
Substitute [tex]\( y \)[/tex] from the first equation into the second equation:
[tex]\[ 2x + 3 = -x + 6 \][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ 2x + 3 = -x + 6 \][/tex]
[tex]\[ 2x + x = 6 - 3 \][/tex]
[tex]\[ 3x = 3 \][/tex]
[tex]\[ x = 1 \][/tex]
Now, substitute [tex]\( x = 1 \)[/tex] back into the first equation to find [tex]\( y \)[/tex]:
[tex]\[ y = 2(1) + 3 = 2 + 3 = 5 \][/tex]
Hence, the coordinates of the point of intersection are [tex]\( (1, 5) \)[/tex].
Answer for (d): The coordinates of the point of intersection are [tex]\( (1, 5) \)[/tex].
