Sure! Let's solve your questions step by step in a simple way.
(a) Write the equation for Sidewalk 1 in slope-intercept form.
Points on Sidewalk 1:
(2, 7) and (0, 0)
Find the slope (m):
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Using points (2, 7) and (0, 0):
\[
m = \frac{7 - 0}{2 - 0} = \frac{7}{2}
\]
Use the slope-intercept form (y = mx + b):
We know the slope (m = 7/2) and use one point (0, 0) to find b:
\[
y = \frac{7}{2}x + b
\]
Substituting (0, 0):
\[
0 = \frac{7}{2}(0) + b
\]
So, \( b = 0 \).
Equation for Sidewalk 1:
\[
y = \frac{7}{2}x
\]
(b) Write the equation for Sidewalk 2 in point-slope form and then in slope-intercept form.
Points on Sidewalk 2:
(3, 1) and (5, 3)
Find the slope (m):
Using the points (3, 1) and (5, 3):
\[
m = \frac{3 - 1}{5 - 3} = \frac{2}{2} = 1
\]
Point-slope form (y - y_1 = m(x - x_1)):
Using point (3, 1):
\[
y - 1 = 1(x - 3)
\]
Slope-intercept form:
Distributing the right side:
\[
y - 1 = x - 3
\]
Adding 1 to both sides:
\[
y = x - 2
\]
(c) Is the system of equations consistent independent, coincident, or inconsistent? Explain.
The equations are:
- Sidewalk 1: \( y = \frac{7}{2}x \)
- Sidewalk 2: \( y = x - 2 \)
These lines have different slopes \( (\frac{7}{2} \text{ vs } 1) \). Different slopes means they will cross at one point, so:
Answer: The system is consistent independent because they intersect at one point.
(d) If the two sidewalks intersect, what are the coordinates of the point of intersection? Use the substitution method and show your work.
Set the two equations equal to each other:
\[ \frac{7}{2}x = x - 2 \]
Solve for x:
First, eliminate \(x\) from the right:
\[
\frac{7}{2}x - x = -2
\]
Convert \(x\) to \( \frac{2}{2}x\):
\[
\frac{7}{2}x - \frac{2}{2}x = -2
\]
Combine like terms:
\[
\frac{5}{2}x = -2
\]
Now, multiply both sides by \( \frac{2}{5} \):
\[
x = -2 \times \frac{2}{5} = -\frac{4}{5}
\]
Plug x back in to find y:
Using Sidewalk 2 equation:
\[
y = -\frac{4}{5} - 2
\]
Convert -2 to a fraction:
\[
y = -\frac{4}{5} - \frac{10}{5} = -\frac{14}{5}
\]
Coordinates of intersection:
The point of intersection is:
\[ \left(-\frac{4}{5}, -\frac{14}{5}\right) \]
So, there you go! If you have more questions, feel free to ask!
1 answer