What is the equation of the line that passes through the points (15, 9) and (-2, 9)?
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Since both points have the same y-coordinate, the line is a horizontal line. The equation of a horizontal line is y = b, where b is the y-coordinate. In this case, b = 9, so the equation of the line is y = 9.
What are the slope and the y-intercept of the linear function that is represented by the graph?
On a coordinate plane, a line goes through points (negative 3, 0) and (negative 2, 4).
The slope is 3, and the y-intercept is 9.
The slope is 3, and the y-intercept is 12.
The slope is 4, and the y-intercept is 9.
The slope is 4, and the y-intercept is 12.
The slope of a line can be found using the formula m = (y2 - y1)/(x2 - x1), where (x1, y1) and (x2, y2) are any two points on the line. Using the given points, we have:
m = (4 - 0)/(-2 - (-3)) = 4/1 = 4
Therefore, the slope of the linear function is 4.
To find the y-intercept, we can use the point-slope form of a linear equation: y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope. Using the point (−3, 0) and the slope 4, we have:
y - 0 = 4(x - (-3))
y = 4x + 12
Therefore, the y-intercept is 12.
The answer is: The slope is 4, and the y-intercept is 12.
A linear function has a slope of Negative StartFraction 7 Over 9 EndFraction and a y-intercept of 3. How does this function compare to the linear function that is represented by the equation y + 11 = Negative StartFraction 7 Over 9 EndFraction (x minus 18)?
It has the same slope and the same y-intercept.
It has the same slope and a different y-intercept.
It has the same y-intercept and a different slope.
It has a different slope and a different y-intercept.
The linear function with a slope of Negative StartFraction 7 Over 9 EndFraction and a y-intercept of 3 can be written as y = Negative StartFraction 7 Over 9 EndFraction x + 3.
The linear function represented by the equation y + 11 = Negative StartFraction 7 Over 9 EndFraction (x minus 18) can be rewritten in slope-intercept form as y = Negative StartFraction 7 Over 9 EndFraction x + 11 + Negative StartFraction 7 Over 9 EndFraction (18) = Negative StartFraction 7 Over 9 EndFraction x + 25.
Comparing the two functions, we see that they have the same slope, Negative StartFraction 7 Over 9 EndFraction, but different y-intercepts: 3 for the first function, and 25 for the second function.
Therefore, the answer is: It has the same slope and a different y-intercept.
What can be concluded about the line represented in the table? Select 3 options.
x
y
–6
–7
2
–3
8
0
The slope is 2.
The slope is One-half.
The y-intercept is –4.
The y-intercept is 8.
The points (–2, –5) and (8, 0) are also on the line.
The points (–5, –2) and (1, 10) are also on the line.
From the given table, we can calculate the slope (m) of the line by using any two points. Using (–6, –7) and (2, –3), we get:
m = (y2 - y1)/(x2 - x1) = (-3 - (-7))/(2 - (-6)) = 4/8 = 1/2
Therefore, option 2, "The slope is One-half," is correct.
We can also calculate the y-intercept (b) of the line by using any point and the slope. Using the point (2, –3) and the slope 1/2, we get:
y - y1 = m(x - x1)
y - (-3) = 1/2(x - 2)
y = 1/2x - 2 + (-3)
y = 1/2x - 5
Therefore, option 3, "The y-intercept is –4," is incorrect, and option 4, "The y-intercept is 8," is also incorrect.
To check whether the points (–2, –5) and (8, 0) are on the line, we can substitute their x and y values into the equation of the line and see if they satisfy it.
For (-2, -5):
y = 1/2x - 5
y = 1/2(-2) - 5
y = -6
The point (-2, -5) does not lie on the line. So, option 5 is incorrect.
For (8, 0):
y = 1/2x - 5
y = 1/2(8) - 5
y = -1
The point (8, 0) does not lie on the line. So, option 5 is also incorrect.
Therefore, the only correct option is:
The slope is One-half.
The graph shows the amount of water that remains in a barrel after it begins to leak. The variable x represents the number of days that have passed since the barrel was filled, and y represents the number of gallons of water that remain in the barrel.
A graph titled Water Supply with number of days on the x-axis and gallons of water left on the y-axis. A line goes through points (6, 33) and (15, 15).
What does the y-intercept represent?
the amount of water that is added to the barrel each day
the amount of water that remains in the water barrel
the amount of water that is initially in the water barrel
the amount that the water barrel decreases each day
The y-intercept of the line is the value of y when x is equal to 0. In this case, we are not given the value of y when x is equal to 0, since the line only goes through the points (6, 33) and (15, 15).
Therefore, we cannot determine what the y-intercept represents based on the information given in the graph.
Option 1, "the amount of water that is added to the barrel each day," is incorrect, as the graph does not provide any information about water being added to the barrel.
Option 2, "the amount of water that remains in the water barrel," is also incorrect, as we have already established that the y-intercept cannot be determined from the given information.
Option 3, "the amount of water that is initially in the water barrel," is a possibility, as the y-intercept could represent the initial amount of water in the barrel before any days have passed. However, we cannot confirm this without knowing the value of y when x is equal to 0.
Option 4, "the amount that the water barrel decreases each day," is also incorrect, as the line in the graph shows the amount of water remaining in the barrel, not the amount by which it is decreasing each day.
Therefore, the answer is: Not enough information is given to determine what the y-intercept represents.
The slope of the graph of the equation y = 2 x minus 2 is 2. What is the y-intercept?
y-intercept =
The y-intercept is the value of y when x is equal to 0. Therefore, we can substitute x = 0 into the equation y = 2x - 2 to find the y-intercept:
y = 2(0) - 2
y = -2
Therefore, the y-intercept is -2.
The cost of a service call to fix a washing machine can be expressed by the linear function y = 45 x + 35, where y represents the total cost and x represents the number of hours it takes to fix the machine. What does the y-intercept represent?
the cost for each hour it takes to repair the machine
the service fee for coming to look at the machine
the total cost for fixing the washing machine
the amount of time that it takes to arrive at the home to make the repairs
In the given linear function y = 45x + 35, the coefficient of x is 45, which represents the cost per hour of repairing the washing machine. The y-intercept, however, is represented by the constant term of the function, which is 35.
Therefore, the y-intercept of the function y = 45x + 35 represents the service fee for coming to look at the machine, i.e., the fixed cost that the repair person charges just for showing up, regardless of how long it takes to make the actual repairs.
Therefore, the answer is: the service fee for coming to look at the machine.
What are the slope and the y-intercept of the linear function that is represented by the table?
x
y
Negative three-fourths
Negative StartFraction 1 Over 30 EndFraction
Negative one-half
Negative StartFraction 2 Over 15 EndFraction
One-fourth
Negative StartFraction 13 Over 30 EndFraction
Two-thirds
Negative three-fifths
The slope is Negative two-fifths, and the y-intercept is Negative one-third.
The slope is Negative one-third, and the y-intercept is Negative two-fifths.
The slope is One-third, and the y-intercept is Negative two-fifths.
The slope is Two-fifths, and the y-intercept is Negative one-third.
To find the slope of the linear function from the given table, we can use any two points and apply the formula:
m = (y2 - y1)/(x2 - x1)
Let's choose the points (-3/4, -1/30) and (1/4, -13/30):
m = (-13/30 - (-1/30))/(1/4 - (-3/4))
m = (-12/30)/(1)
m = -2/5
Therefore, the slope of the linear function is Negative two-fifths.
To find the y-intercept of the linear function, we can use any point on the line and the slope. Let's use the point (-1/2, 2/15):
y = mx + b
2/15 = (-2/5)(-1/2) + b
2/15 = 1/5 + b
b = 2/15 - 1/5
b = -1/30
Therefore, the y-intercept of the linear function is Negative one-third.
Therefore, the answer is: The slope is Negative two-fifths, and the y-intercept is Negative one-third.
Which linear function has the same slope as the one that is represented by the table?
x
y
Negative one-half
One-fifth
Negative one-fifth
StartFraction 7 Over 50 EndFraction
One-fifth
StartFraction 3 Over 50 EndFraction
One-half
0
y = negative one-half x + StartFraction 1 Over 10 EndFraction
y = negative one-fifth x + one-half
y = one-fifth x minus one-half
y = one-half x minus StartFraction 1 Over 10 EndFraction
To find the slope of the linear function from the given table, we can use any two points and apply the formula:
m = (y2 - y1)/(x2 - x1)
Let's choose the points (-1/2, 1/5) and (1/2, 0):
m = (0 - 1/5)/(1/2 - (-1/2))
m = (-1/5)/(1)
m = -1/5
Therefore, the slope of the linear function represented by the table is negative one-fifth.
The answer choice that has the same slope is:
y = negative one-fifth x + one-half
Therefore, the answer is: y = negative one-fifth x + one-half.
Which graph represents a linear function that has a slope of 0.5 and a y-intercept of 2?
On a coordinate plane, a line goes through points (negative 2, 0) and (0, 1).
On a coordinate plane, a line goes through points (0, 2) and (4, 0).
On a coordinate plane, a line goes through points (0, 2) and (2, 3).
On a coordinate plane, a line goes through points (negative 4, 0) and (0, 2).
We can determine the equation for the linear function from the slope-intercept form:
y = mx + b
where m is the slope and b is the y-intercept.
Substituting values for m and b, we get:
y = 0.5x + 2
Therefore, we are looking for a graph where the line passes through the point (0, 2), and has a slope of 0.5.
Option 2 is not correct, as it has a y-intercept of 2, but the slope is -0.5, not 0.5.
Option 3 is also not correct, as it has a y-intercept of 2, but the slope is greater than 0.5.
Option 4 is also not correct, as it has a slope of 0.5, but the y-intercept is not 2.
Therefore, the answer is option 1, where the line passes through the points (-2, 0) and (0, 1). Since this line has a slope of 0.5 and passes through the point (0,2), which is the y-intercept, it represents the linear function y = 0.5x + 2.
Brooke found the equation of the line passing through the points (–7, 25) and (–4, 13) in slope-intercept form as follows.
Step 1: m = StartFraction 13 minus 25 Over negative 4 minus (negative 7) EndFraction = StartFraction negative 12 Over 3 EndFraction = negative 4. Step 2: y = negative 4 x + b. 25 = negative 4 (negative 7) + b. 25 = 28 + b. 25 minus 28 = 28 + b minus 28. b = negative 3. Step 3: y = negative 3 x minus 4
What was Brooke’s error?
She found the incorrect slope in step 1.
She mixed up the x- and y-coordinates when she plugged in the point in step 2.
She found the incorrect y-intercept in step 2.
She mixed up the slope and y-intercept when she wrote the equation in step 3.
In step 2 of Brooke's work, she used the correct slope of -4, but plugged in the wrong coordinates for one of the points. She used the point (-7, 25), which is correct, but when plugging in the other point, she used (-4, 13) instead of (–4, 25).
Using the correct point (–4, 25), we have:
25 = -4(-4) + b
25 = 16 + b
b = 9
Therefore, the correct equation for the line passing through the points (–7, 25) and (–4, 25) is:
y = -4x + 9
Therefore, the answer is: Brooke mixed up the x- and y-coordinates when she plugged in the point in step 2.