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:** \( (9, -\frac{2}{3}) \)