Interactive multiple-choice quiz: Creating Linear Equations ريفيل
This document contains exercises on finding the y-intercept and writing the equation of a line using the slope-intercept form $y = mx + b$. These problems help students practice substituting points and slopes into the linear equation formula to derive the complete equation of a line.
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Using the slope-intercept form y = mx + b, find the value of b for the line that passes through (-8, 6) with a slope of $m = -\frac{3}{4}$
Explanation
Substitute the point (-8, 6) and the slope $m = -\frac{3}{4}$ into the slope-intercept equation y = mx + b. This gives $6 = -\frac{3}{4}(-8) + b$, which simplifies to 6 = 6 + b. Subtracting 6 from both sides, we find b = 0.
Question 2
Points: 1
Write the final equation of the line in slope-intercept form using the slope $m = -\frac{3}{4}$ and the y-intercept b = 0
Explanation
The slope-intercept form is y = mx + b. Plugging in the given values $m = -\frac{3}{4}$ and b = 0, the equation becomes $y = -\frac{3}{4}x + 0$, which simplifies to $y = -\frac{3}{4}x$.
Question 3
Points: 10
The total monthly cost of Ayzha's gym membership increases by 5 per class she attends. After signing up for 4 classes one month, her total cost is49.99. Which equation represents Ayzha's total monthly cost y after attending x classes?
Explanation
The slope is 5. Substitute (4, 49.99) into y = 5x + b: b = 49.99 - 20 = 29.99, so y = 5x + 29.99.
Question 4
Points: 10
Write an equation of the line that passes through (-5, -3) and (-7, -12).
Explanation
m = (-12 - (-3))/(-7 - (-5)) = -9/(-2) = 9/2. Then b = -3 - (9/2)(-5) = 39/2.
Question 5
Points: 10
The table shows the number of students enrolled in public high schools in the United States since 2010. Which linear equation best represents the data if x represents the number of years since 2010?
Explanation
Using the first and last table entries, the average rate is (14912 - 14749)/(2015 - 2011) = 40.75 thousand students per year. Since 2011 corresponds to x = 1, b = 14749 - 40.75 = 14708.25. This is an approximate endpoint model, not an exact fit to every row.
Question 6
Points: 10
Write an equation of the line that passes through (4, 2) and has slope 1/2.
Explanation
With m = 1/2, b = 2 - (1/2)(4) = 0, so y = (1/2)x.
Question 7
Points: 10
Write an equation of the line that passes through (3, -2) and has slope 1/3.
Explanation
b = -2 - (1/3)(3) = -3, so y = (1/3)x - 3.
Question 8
Points: 10
Write an equation of the line that passes through (6, 4) and has slope -3/4.
Explanation
b = 4 - (-3/4)(6) = 4 + 4.5 = 8.5.
Question 9
Points: 10
Write an equation of the line that passes through (4, 3) and has slope 1/2.
Explanation
b = 3 - (1/2)(4) = 1, so y = (1/2)x + 1.
Question 10
Points: 10
Write an equation of the line that passes through (1, -5) and has slope -3/2.
Explanation
b = -5 - (-3/2)(1) = -3.5, so y = -(3/2)x - 3.5.
Question 11
Points: 10
Determine the equation in point-slope form for the line that passes through (7, 5) with a slope of -3.
Explanation
Substitute x1 = 7, y1 = 5 and m = -3 into y - y1 = m(x - x1).
Question 12
Points: 10
Select an equation in point-slope form for the line that passes through (-16, 18) and (-11, -2).
Explanation
The slope is (-2 - 18)/(-11 - (-16)) = -20/5 = -4. Using (-11, -2), the equation is y + 2 = -4(x + 11).
Question 13
Points: 10
Write y + 3 = -(1/2)(x - 8) in slope-intercept form.
Explanation
Distribute -1/2 to get y + 3 = -(1/2)x + 4. Subtract 3: y = -(1/2)x + 1.
Question 14
Points: 10
The provided table details the relationship between distance and total fare cost for taxi rides, where x denotes distance in miles and y denotes cost in dollars. Which of the following equations correctly represent the situation in point-slope form? A: y - 13.4 = 2.6(x - 4); B: y - 22.5 = 2.6(x - 7.5); C: y = 2.6x + 3; D: y - 6.9 = (5/13)(x - 1.5); F: y + 34.85 = 2.6(x + 12.25). Select the correct group.
Explanation
The rate is (13.40 - 6.90)/(4 - 1.5) = 2.6 dollars per mile. A and B use this slope and points from the table. C represents the same line but is in slope-intercept form, not point-slope form. D has the wrong slope; F has incorrect signs.
Question 15
Points: 10
Write y = -(7/2)x + 5 in standard form Ax + By = C, using integer coefficients with A > 0.
Explanation
Multiply by 2: 2y = -7x + 10. Move 7x to the left: 7x + 2y = 10. The equivalent equation in B is excluded by the stated positive-A convention.
Question 16
Points: 10
Select the equation in standard form for the line that passes through (-9, 8) and (1, -12).
Explanation
m = (-12 - 8)/(1 - (-9)) = -2. Using (1, -12), b = -10. Thus y = -2x - 10, or 2x + y = -10.
Question 17
Points: 10
Write an equation for the line that passes through (8, 2) and is parallel to the graph of y = (3/4)x + 4.
Explanation
Parallel lines have the same slope, 3/4. Then b = 2 - (3/4)(8) = -4.
Question 18
Points: 10
Select the equation in slope-intercept form for the line that passes through (5, 0) and is perpendicular to the graph of x - 6y = 1.
Explanation
The given line has slope 1/6, so the perpendicular slope is -6. Passing through (5, 0) gives b = 30. A is in the requested slope-intercept form; B is equivalent but is in standard form.
Question 19
Points: 10
Determine whether lines CD and KL are parallel, perpendicular, or neither for C(4, 10), D(-1, 12), K(6, -5), and L(1, -3).
Explanation
Slope CD = (12 - 10)/(-1 - 4) = -2/5. Slope KL = (-3 - (-5))/(1 - 6) = -2/5. Their y-intercepts differ, so the lines are parallel.
Question 20
Points: 10
Complete the sentence: When a = 4 and b = 4 for y = ax - 5 and y = bx + 3, the graphs are:
Explanation
The slopes are both 4 and the y-intercepts are different (-5 and 3), so the lines are parallel.
Question 21
Points: 10
Complete the sentence: When a = -3 and b = 5 for y = ax - 5 and y = bx + 3, the graphs are:
Explanation
The slopes are different, and their product is -15, not -1. The lines are neither parallel nor perpendicular.
Question 22
Points: 10
Complete the sentence: When a = -2 and b = 1/2 for y = ax - 5 and y = bx + 3, the graphs are:
Explanation
The product of the slopes is (-2)(1/2) = -1, so the lines are perpendicular.
Question 23
Points: 10
Find the slope of the line that passes through the given points (4, 3) and (-1, 6).
Explanation
m = (6 - 3)/(-1 - 4) = 3/(-5) = -3/5.
Question 24
Points: 10
Write an equation in point-slope form for the line that passes through (-7, 6) with slope m = 0.
Explanation
Substitute x1 = -7, y1 = 6 and m = 0 into y - y1 = m(x - x1).
Question 25
Points: 10
Write an equation in point-slope form for the line that passes through (-2, 11) with slope m = 4/3.
Explanation
Use y - y1 = m(x - x1): y - 11 = (4/3)(x - (-2)) = (4/3)(x + 2).
Question 26
Points: 10
Determine the inverse of the relation: {(-1.4, 5), (1.3, 6.5), (3, -8), (3.05, 9)}.
Explanation
Find the inverse by swapping the x- and y-coordinates in every ordered pair.
Question 27
Points: 10
Find the inverse of the relation shown in the table.
Explanation
The original pairs are (-11, 9), (0.3, -2), (-3, -8) and (3.5, 2). Swapping their coordinates gives {(9, -11), (-2, 0.3), (-8, -3), (2, 3.5)}.
Question 28
Points: 10
Graph the inverse of the relation. The graph of the relation passes through the points (-1, -5), (0, -2), (1, 1), and (2, 4). Which set of points represents the inverse relation?
Explanation
Swap the coordinates of each original point: (-1, -5) becomes (-5, -1), (0, -2) becomes (-2, 0), (1, 1) stays (1, 1), and (2, 4) becomes (4, 2).
Question 29
Points: 10
Find the inverse of f(x) = 5x + 10.
Explanation
Replace f(x) with y, swap x and y, then solve: x = 5y + 10 gives y = (x - 10)/5 = (1/5)x - 2.
Question 30
Points: 10
Find the inverse of f(x) = -(2/3)x - 8.
Explanation
Swap x and y: x = -(2/3)y - 8. Then x + 8 = -(2/3)y, so y = -(3/2)x - 12.
Question 31
Points: 10
CANDLES: Javi is making candles to sell at an upcoming festival. He has already made 38 candles, and he makes 24 candles each day. The function C(x) = 24x + 38 represents the total number of candles in inventory, where x is the number of days since he began making more candles. Part A: Select the inverse of C(x) = 24x + 38.
Explanation
Swap x and y and solve: x = 24y + 38 gives y = (x - 38)/24 = (1/24)x - 19/12.
Question 32
Points: 10
Part B: Javi already has 38 candles and makes 24 more per day, so C(x) = 24x + 38. How many days will it take him to have a total of 350 candles?
Explanation
Solve 24x + 38 = 350: x = (350 - 38)/24 = 312/24 = 13 days exactly.
Question 33
Points: 10
Find the inverse of the relation.
Explanation
Swap x and y in each table row: (-9, -1) becomes (-1, -9), and similarly for the remaining rows.
Question 34
Points: 10
Find the inverse of the relation.
Explanation
The inverse is obtained by swapping coordinates: (1, 8) becomes (8, 1), (2, 6) becomes (6, 2), (3, 4) becomes (4, 3), (4, 2) becomes (2, 4), and (5, 0) becomes (0, 5).
Question 35
Points: 10
Find the inverse of the relation.
Explanation
Swap the coordinates of every pair. In particular, (0, 1) becomes (1, 0) and (2, 0) becomes (0, 2).
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