I'll be happy to help you with each of these problems. Here are the step-by-step solutions: **1. Write the equation of a line given the following: m= 6 (4,-1)** To write the equation of a line, we can use the point-slope form: y - y1 = m(x - x1),
Here are the step-by-step solutions to the problems provided:
### 1. Write the equation of a line given \( m = 6 \) and the point \( (4, -1) \)
Using the point-slope form \( y - y_1 = m(x - x_1) \):
\[
y - (-1) = 6(x - 4)
\]
\[
y + 1 = 6x - 24
\]
\[
y = 6x - 25
\]
**Equation:** \( y = 6x - 25 \)
---
### 2. Graph the line \( y < 7x + 4 \)
- First, graph the line \( y = 7x + 4 \) (the boundary line).
- This line has a slope of 7 and a y-intercept of 4.
- Since the inequality is "<", shade below the line.
**Note:** The actual graph cannot be displayed here, but ensure you use a dashed line for the boundary to indicate the inequality.
---
### 3. Is the following a function? \( (3,4), (6,2), (7,9), (8,1) \)
Each x-value is unique:
- **x-values:** 3, 6, 7, 8
- **y-values:** 4, 2, 9, 1
Since there are no repeated x-values, **Yes, this is a function.**
---
### 4. Find the solution set for the system using substitution: \( 7x + 2y = -50, x + 4y = 4 \)
1. Solve the second equation for \( x \):
\[
x = 4 - 4y
\]
2. Substitute into the first equation:
\[
7(4 - 4y) + 2y = -50
\]
\[
28 - 28y + 2y = -50
\]
\[
-26y = -78 \implies y = 3
\]
3. Substitute \( y = 3 \) back to find \( x \):
\[
x + 4(3) = 4 \implies x + 12 = 4 \implies x = -8
\]
**Solution Set:** \( (-8, 3) \)
---
### 5. Graph the solution set for the system: \( x + 2y < 3, y < 6 \)
1. Graph \( x + 2y = 3 \) (boundary line) and shade below for \( x + 2y < 3 \).
2. Graph \( y = 6 \) (horizontal line) and shade below for \( y < 6 \).
**Note:** The intersection of the shaded regions represents the solution set.
---
### 6. Calculate total miles Mary will walk each week
1. Normal walking distance:
\[
6 \text{ miles/day} \times 7 \text{ days} = 42 \text{ miles/week}
\]
2. Increased distance for 3 days:
\[
12 \text{ miles/day} \times 3 \text{ days} = 36 \text{ miles}
\]
3. Adjust for the original distance walked on those 3 days:
\[
18 \text{ miles} = 6 \text{ miles/day} \times 3 \text{ days}
\]
4. Total miles:
\[
42 + 36 - 18 = 60 \text{ miles/week}
\]
**Total miles per week:** \( 60 \)
---
### 7. Find the equation of a line that passes through \( (2, -3) \) and \( (1, 6) \)
1. Calculate the slope \( m \):
\[
m = \frac{6 - (-3)}{1 - 2} = \frac{9}{-1} = -9
\]
2. Use point-slope form with point \( (2, -3) \):
\[
-3 = -9(2) + b \implies -3 = -18 + b \implies b = 15
\]
**Equation:** \( y = -9x + 15 \)
---
### 8. Create a graph showing a function and a graph that is not a function
**Function:** A straight line (e.g., \( y = 2x + 1 \)).
**Not a Function:** A circle (e.g., \( x^2 + y^2 = 9 \)).
**Note:** Actual graphs cannot be displayed.
---
### 9. Graph the solution set for the system: \( x + 4y \leq -20, y < 3x \)
1. Graph \( x + 4y = -20 \) (and shade below).
2. Graph \( y = 3x \) (and shade below).
**Note:** The intersection of the shaded regions represents the solution set.
---
### 10. Find the solution set for the system using elimination: \( 2x + 3y = 16, x - 3y = 11 \)
1. Add the equations to eliminate \( y \):
\[
(2x + 3y) + (x - 3y) = 16 + 11
\]
\[
3x = 27 \implies x = 9
\]
2. Substitute \( x = 9 \) back into one equation:
\[
2(9) + 3y = 16
\]
\[
18 + 3y = 16 \implies 3y = -2 \implies y = -\frac{2}{3}
\]
**Solution Set:** \( (9, -\frac{2}{3}) \)