Find the equation of a line that has the indicated properties and sketch the line. a passing through (−1, 3) and parallel to y = 4x − 5

here are the solutions to your questions. 3. John paid 7.5% sales tax and 150 title and license fee when he bought a new car for a total of 25868.5. Find the purchased price of a car? Step 1: Let P be the purchased price of the car. The sales tax is 7.5\% of P, which is 0.075P. The total cost is the sum of the purchased price, sales tax, and license fee. P + 0.075P + 150 = 25868.5 Step 2: Combine the terms involving P. 1.075P + 150 = 25868.5 Step 3: Subtract the license fee from the total cost. 1.075P = 25868.5 - 150 1.075P = 25718.5 Step 4: Divide by 1.075 to find the purchased price P. P = (25718.5)/(1.075) P = \23924.19 4. Find the slope of the straight line passing through the indicated points: The slope m between two points (x_1, y_1) and (x_2, y_2) is given by the formula m = (y_2 - y_1)/(x_2 - x_1). i) (4, 1), (7, 5) Step 1: Substitute the coordinates into the slope formula. m = (5 - 1)/(7 - 4) Step 2: Calculate the slope. m = (4)/(3) m = (4)/(3) ii) (5, -3), (6, -4) Step 1: Substitute the coordinates into the slope formula. m = (-4 - (-3))/(6 - 5) Step 2: Calculate the slope. m = (-4 + 3)/(1) = (-1)/(1) m = -1 iii) (1, 0), (0, 5) Step 1: Substitute the coordinates into the slope formula. m = (5 - 0)/(0 - 1) Step 2: Calculate the slope. m = (5)/(-1) m = -5 iv) (2, -4), (3, -4) Step 1: Substitute the coordinates into the slope formula. m = (-4 - (-4))/(3 - 2) Step 2: Calculate the slope. m = (-4 + 4)/(1) = (0)/(1) m = 0 5. Find the equation of a line that has the indicated properties and sketch the line: To sketch the line, plot the two points used to derive the equation and draw a straight line through them. a) passing through (-1, 3) and parallel to y = 4x - 5 Step 1: Parallel lines have the same slope. The slope of y = 4x - 5 is m = 4. Step 2: Use the point-slope form y - y_1 = m(x - x_1) with point (-1, 3) and m=4. y - 3 = 4(x - (-1)) y - 3 = 4(x + 1) y - 3 = 4x + 4 Step 3: Solve for y to get the slope-intercept form. y = 4x + 7 y = 4x + 7 To sketch, plot points like (0, 7) and (-1, 3) and draw the line. b) has slope 0 and y-intercept -(1)/(2) Step 1: Use the slope-intercept form y = mx + b. Given m = 0 and b = -(1)/(2). y = 0x - (1)/(2) Step 2: Simplify the equation. y = -(1)/(2) y = -(1)/(2) To sketch, plot points like (0, -(1)/(2)) and (1, -(1)/(2)) and draw the horizontal line. c) passes through (3, -1) and (-2, -9) Step 1: Find the slope m using the two points. m = (-9 - (-1))/(-2 - 3) = (-8)/(-5) = (8)/(5) Step 2: Use the point-slope form y - y_1 = m(x - x_1) with point (3, -1) and m=(8)/(5). y - (-1) = (8)/(5)(x - 3) y + 1 = (8)/(5)x - (24)/(5) Step 3: Solve for y. y = (8)/(5)x - (24)/(5) - 1 y = (8)/(5)x - (24)/(5) - (5)/(5) y = (8)/(5)x - (29)/(5) y = (8)/(5)x - (29)/(5) To sketch, plot points (3, -1) and (-2, -9) and draw the line. d) passing through origin and has slope -5 Step 1: The origin is the point (0, 0). Use the slope-intercept form y = mx + b. Given m = -5 and since it passes through (0, 0), the y-intercept b = 0. y = -5x + 0 Step 2: Simplify the equation. y = -5x y = -5x To sketch, plot points like (0, 0) and (1, -5) and draw the line. e) passing through (-5, 4) and perpendicular to the line 2y = x + 1 Step 1: Find the slope of the given line 2y = x + 1. Divide by 2 to get y = (1)/(2)x + (1)/(2). The slope is m_1 = (1)/(2). Step 2: The slope of a perpendicular line m_2 is the negative reciprocal of m_1. m_2 = -(1)/(m_1) = -(1)/(1/2) = -2 Step 3: Use the point-slope form y - y_1 = m(x - x_1) with point (-5, 4) and m_2 = -2. y - 4 = -2(x - (-5)) y - 4 = -2(x + 5) y - 4 = -2x - 10 Step 4: Solve for y. y = -2x - 6 y = -2x - 6 To sketch, plot points like (-5, 4) and (0, -6) and draw the line. f) perpendicular to y = 3x - 5 and passing through (3, 4) Step 1: The slope of the given line y = 3x - 5 is m_1 = 3. Step 2: The slope of a perpendicular line m_2 is the negative reciprocal of m_1. m_2 = -(1)/(3) Step 3: Use the point-slope form y - y_1 = m(x - x_1) with point (3, 4) and m_2 = -(1)/(3). y - 4 = -(1)/(3)(x - 3) y - 4 = -(1)/(3)x + 1 Step 4: Solve for y. y = -(1)/(3)x + 5 y = -(1)/(3)x + 5 To sketch, plot points like (3, 4) and (0, 5) and draw the line. g) passing through (2, -8) and parallel to x = -4 Step 1: The line x = -4 is a vertical line. A line parallel to a vertical line is also a vertical line. Step 2: The equation of a vertical line passing through a point (x_1, y_1) is x = x_1. For the point (2, -8), the equation is x = 2. x = 2 To sketch, plot points like (2, -8) and (2, 0) and draw the vertical line. 6. A small company produces chairs. The weekly fixed and variable costs per chair are 1100 and 40 respectively. Find the total weekly cost of manufacturing x chairs. How many chairs can be manufactured for a total weekly cost of 4,500? Part 1: Find the total weekly cost of manufacturing x chairs. Step 1: The total weekly cost C is the sum of the fixed cost and the total variable cost. Fixed cost = 1100 Variable cost per chair = 40 Total variable cost for x chairs = 40x C = 1100 + 40x C = 1100 + 40x Part 2: How many chairs can be manufactured for a total weekly cost of 4,500? Step 1: Set the total cost C to 4500. 4500 = 1100 + 40x Step 2: Subtract the fixed cost from the total cost. 4500 - 1100 = 40x 3400 = 40x Step 3: Divide by 40 to find the number of chairs x. x = (3400)/(40) x = 85 chairs 7. The sales of a company were 33,000 in its third year of operation and 95,000 in its sixth year. If y denote sales in year x, what were the sales in the fifth year? Assume a linear relationship between sales and year. We have two points (x, y): (3, 33000) and (6, 95000). Step 1: Find the slope m of the line. m = (95000 - 33000)/(6 - 3) = (62000)/(3) Step 2: Use the point-slope form y - y_1 = m(x - x_1) with point (3, 33000) and m = (62000)/(3). y - 33000 = (62000)/(3)(x - 3) y - 33000 = (62000)/(3)x - (62000)/(3) × 3 y - 33000 = (62000)/(3)x - 62000 Step 3: Solve for y to get the equation for sales. y = (62000)/(3)x - 62000 + 33000 y = (62000)/(3)x - 29000 Step 4: Substitute x=5 (fifth year) into the equation to find the sales. y = (62000)/(3)(5) - 29000 y = (310000)/(3) - 29000 y = 103333.333... - 29000 y = 74333.333... Sales in the fifth year = \74333.33 Last free one today — make it count tomorrow, or type /upgrade for unlimited.