1. For the data in the table, does y vary directly with x? If it does, write an equation for the direct
variation.
x y
2 5
6 15
18 45
(1 point)
yes; y = 0.4x
yes; y = 10x
yes; y = 2.5x
no; y does not vary directly with x
2. For the data in the table, does y vary directly with x? If it does, write an equation for the direct
variation.
x y
40 32
28 16
16 12
(1 point)
no; y does not vary directly with x
yes; y = 2x
yes; y = 1.5x
yes; y = 0.2x
5. The table shows the number of miles driven over time.
time(hours) Distance(miles)
4 204
6 306
8 408
10 510
Express the relationship between distance and time in simplified form as a unit rate. Determine
which statement correctly interprets this relationship.
(1 point)
51/1; your car travels 51 miles every 1 hour.
204; your car travels 204 miles.
1/51; your car travels 51 miles every 1 hour.
10; your car travels for 10 hours
What is the slope of the line that passes through the pair of points (2, 9), and (5, 12)? (1 point)
0
1
3
undefined
What is the slope of the line that passes through the pair of points (3, –4), and (5, –7)?
2/3
-2/3
-3/2
3/2
Write an equation in point-slope form for the line through the given point with the given slope.
(5, 2); m = 3
(1 point)
y + 2 = 3(x – 5)
y + 2 = 3x – 5
y – 2 = 3(x + 5)
y – 2 = 3(x – 5)
11. Write an equation in point-slope form for the line through the given point with the given slope.
(–4, –2); m = -1/2
(1 point)
y + 2 = -1/2(x – 4)
y + 2 = ( -1/2) x – 4
y + 2 = -1/2(x + 4)
y – 2 = -1/2(x – 4)
The table shows the height of a plant as it grows. Which equation in point-slope form gives the
plant’s height at any time?
Time (months) Plant Height
(cm)
2 16
4 32
6 48
8 64
(1 point)
y – 16 = 8(x – 2)
y-16=16(x+2)
y + 16 = 8(x + 2)
The relationship is nonlinear
4. Write y = –2/3 x + 7 in standard form using integers. (1 point)
2x + 3y = 21
3x – 2y = 21
–2x – 3y = 21
–2x + 3y = 7
Write an equation for the line that is parallel to the given line and passes through the given
point.
y = 3x + 4; (6, 8)
(1 point)
y = 3x – 10
y = 3x + 6
y = –3x + 6
y = – x 1/3 – 6
Tell whether the lines for each pair of equations are parallel, perpendicular, or neither.
y = –3x + 7
–2x + 6y = 3
(1 point)
parallel
perpendicular
neither
17. Tell whether the lines for each pair of equations are parallel, perpendicular, or neither.
y = -1/5 x + 6
–2x + 10y = 5
(1 point)
parallel
perpendicular
neither
18. Write the equation of a line that is perpendicular to the given line and that passes through the
given point.
y-4=5/2(x+3);(-3,4)
y+4=-2/5(x-3)
y-3=-2/5(x+4)
y-4=-2/5(x+3)
y+4=2/5(x-3)
1. yes; y = 2.5x
2. no; y does not vary directly with x
3. 51/1; your car travels 51 miles every 1 hour.
4. 3/3 = 1
5. -3/2
6. y - 2 = 3(x - 5)
7. y + 2 = -1/2(x - 4)
8. y - 16 = 8(x - 2)
9. 2x + 3y = 21
10. y = 3x - 10
11. perpendicular
12. neither
13. y + 4 = -2/5(x - 3)
1. To determine if y varies directly with x, we need to check if the ratio between y and x remains constant for all the given pairs of values. Let's calculate the ratios for each pair:
For (2, 5): y/x = 5/2 = 2.5
For (6, 15): y/x = 15/6 = 2.5
For (18, 45): y/x = 45/18 = 2.5
Since the ratio y/x is the same for all pairs, y does vary directly with x. Therefore, the correct equation for direct variation is y = 2.5x.
2. Let's calculate the ratios for the given pairs:
For (40, 32): y/x = 32/40 = 0.8
For (28, 16): y/x = 16/28 = 0.57
For (16, 12): y/x = 12/16 = 0.75
Since the ratios are not the same for all pairs, y does not vary directly with x.
5. To express the relationship between distance and time as a unit rate, we need to divide the distance by the time:
For (4, 204): distance/time = 204/4 = 51
For (6, 306): distance/time = 306/6 = 51
For (8, 408): distance/time = 408/8 = 51
For (10, 510): distance/time = 510/10 = 51
The unit rate is 51 miles per hour. The correct statement interpreting this relationship is: 51/1; your car travels 51 miles every 1 hour.
The slope of a line passing through two points (x1, y1) and (x2, y2) can be calculated using the formula: slope = (y2 - y1) / (x2 - x1)
For (2, 9) and (5, 12): slope = (12 - 9) / (5 - 2) = 3 / 3 = 1
Therefore, the slope of the line is 1.
For (3, -4) and (5, -7): slope = (-7 - (-4)) / (5 - 3) = (-7 + 4) / 2 = -3 / 2
Therefore, the slope of the line is -3/2.
To write an equation in point-slope form, we use the formula: y - y1 = m(x - x1)
For (5, 2) and m = 3: y - 2 = 3(x - 5)
Therefore, the equation in point-slope form is: y - 2 = 3(x - 5).
For (-4, -2) and m = -1/2: y - (-2) = -1/2(x - (-4))
Simplifying, we get: y + 2 = -1/2(x + 4)
Therefore, the equation in point-slope form is: y + 2 = -1/2(x + 4).
The table shows a linear relationship between time and plant height. To find the equation, we can use the formula: y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.
Using (2, 16) and (8, 64), we can calculate the slope: m = (64 - 16) / (8 - 2) = 48 / 6 = 8
Therefore, the equation in point-slope form is: y - 16 = 8(x - 2).
To write the equation y = -2/3x + 7 in standard form, we can multiply by 3 to get rid of the fraction:
3y = -2x + 21
Then, we rearrange the equation to have the variables on the left side and the constant term on the right:
2x + 3y = 21
Therefore, the equation in standard form is: 2x + 3y = 21.
To find the equation of a line parallel to y = 3x + 4 and passing through (6, 8), we know that parallel lines have the same slope. Therefore, the equation of the line will also be y = 3x + b, where b is the y-intercept.
Using the point (6, 8), we can substitute the values and solve for b:
8 = 3(6) + b
8 = 18 + b
b = -10
Therefore, the equation of the line is: y = 3x - 10.
For the pair of equations:
y = -3x + 7
-2x + 6y = 3
We can rewrite the second equation in slope-intercept form:
6y = 2x + 3
y = (2/6)x + 1/2
y = (1/3)x + 1/2
Since the slopes of the two lines are not the same and their product is not -1, the lines are neither parallel nor perpendicular.
For the pair of equations:
y = -1/5x + 6
-2x + 10y = 5
We can rewrite the second equation in slope-intercept form:
10y = 2x + 5
y = (2/10)x + 1/2
y = (1/5)x + 1/2
Since the slopes of the two lines are negative reciprocals (the product of their slopes is -1), the lines are perpendicular.
To find the equation of a line perpendicular to y - 4 = (5/2)(x + 3) and passing through (-3, 4), we first need to find the negative reciprocal of the slope (5/2) to get the slope of the perpendicular line, which is -2/5.
Using the point-slope form, we can write the equation:
y - 4 = (-2/5)(x - (-3))
y - 4 = (-2/5)(x + 3)
y - 4 = (-2/5)x - 6/5
y = (-2/5)x - 6/5 + 4
y = (-2/5)x - 6/5 + 20/5
y = (-2/5)x + 14/5
Therefore, the equation of the line is: y = (-2/5)x + 14/5.
1. yes; y = 2.5x
2. no; y does not vary directly with x
5. 51/1; your car travels 51 miles every 1 hour.
Slope is 1
Slope is -3/2
Point-slope form equation: y + 2 = 3(x – 5)
Point-slope form equation: y + 2 = -1/2(x – 4)
The relationship is nonlinear
4. 3x - 2y = 21
Parallel line equation: y = 3x + 6
Lines are neither parallel nor perpendicular
Lines are perpendicular
Perpendicular line equation: y - 3 = -2/5(x + 4)