Question
Write the linear equation in point-slope form given a line that passes through the point (−14,8) and has a slope of −3 .(1 point)
Responses
y+8=−3(x+14)
y plus 8 equals negative 3 left parenthesis x plus 14 right parenthesis
y−8=−3(x+14)
y minus 8 equals negative 3 left parenthesis x plus 14 right parenthesis
y+8=3(x+14)
y plus 8 equals 3 left parenthesis x plus 14 right parenthesis
y−8=−3(x−14)
If a line can be represented by the equation y+2=13(x−1/2) , which of the following sets of points lies on the line?(1 point)
Responses///
(12,−2) and (15,−1)
left parenthesis 12 comma negative 2 right parenthesis and left parenthesis 15 comma negative 1 right parenthesis
(12,−2) and (13,1)
left parenthesis 12 comma negative 2 right parenthesis and left parenthesis 13 comma 1 right parenthesis
(−12,2) and (−9,3)
left parenthesis negative 12 comma 2 right parenthesis and left parenthesis negative 9 comma 3 right parenthesis
(−12,2) and (−11,3)
Tamara is analyzing a line that passes through the point (−5,3) and has a slope of −1 . How should she write the equation of the line in slope-intercept form?(1 point)
Responses
y=x−1
y equals x minus 1
y=−x−2
y equals negative x minus 2
y=−x+3
y equals negative x plus 3
y=−x−5
A line passes through the point (2,9) and has a slope of 1/2 . Which option correctly writes the equation of the line in standard form?(1 point)
Responses
2x−2y=−16
2 x minus 2 y equals negative 16
x−2y=−16
x minus 2 y equals negative 16
−1/2x+y=8
negative Start Fraction 1 over 2 End Fraction x plus y equals 8
y=1/2x+8
Rahul is presented with the equation of a line in slope-intercept form: y=2/3x+1/4 . By what number can he multiply the equation in order to clear the fractions and rewrite the equation in standard form?(1 point)
Responses
12
12
4
4
6
6
3
if you dont know take your best guess
Responses
y+8=−3(x+14)
y plus 8 equals negative 3 left parenthesis x plus 14 right parenthesis
y−8=−3(x+14)
y minus 8 equals negative 3 left parenthesis x plus 14 right parenthesis
y+8=3(x+14)
y plus 8 equals 3 left parenthesis x plus 14 right parenthesis
y−8=−3(x−14)
If a line can be represented by the equation y+2=13(x−1/2) , which of the following sets of points lies on the line?(1 point)
Responses///
(12,−2) and (15,−1)
left parenthesis 12 comma negative 2 right parenthesis and left parenthesis 15 comma negative 1 right parenthesis
(12,−2) and (13,1)
left parenthesis 12 comma negative 2 right parenthesis and left parenthesis 13 comma 1 right parenthesis
(−12,2) and (−9,3)
left parenthesis negative 12 comma 2 right parenthesis and left parenthesis negative 9 comma 3 right parenthesis
(−12,2) and (−11,3)
Tamara is analyzing a line that passes through the point (−5,3) and has a slope of −1 . How should she write the equation of the line in slope-intercept form?(1 point)
Responses
y=x−1
y equals x minus 1
y=−x−2
y equals negative x minus 2
y=−x+3
y equals negative x plus 3
y=−x−5
A line passes through the point (2,9) and has a slope of 1/2 . Which option correctly writes the equation of the line in standard form?(1 point)
Responses
2x−2y=−16
2 x minus 2 y equals negative 16
x−2y=−16
x minus 2 y equals negative 16
−1/2x+y=8
negative Start Fraction 1 over 2 End Fraction x plus y equals 8
y=1/2x+8
Rahul is presented with the equation of a line in slope-intercept form: y=2/3x+1/4 . By what number can he multiply the equation in order to clear the fractions and rewrite the equation in standard form?(1 point)
Responses
12
12
4
4
6
6
3
if you dont know take your best guess
Answers
Answered by
GPT-4o mini
Let’s go through each question one by one.
1. **Point-Slope Form of the Line**
The point-slope form of a line is given by the equation:
\[
y - y_1 = m(x - x_1)
\]
where \(m\) is the slope and \((x_1, y_1)\) is a point on the line. Given a point \((-14, 8)\) and a slope of \(-3\), we can plug these into the formula:
\[
y - 8 = -3(x + 14)
\]
So the correct response is:
\[
\text{y + 8 = -3(x + 14)}
\]
2. **Identifying Points on the Line**
To verify which of the given points lies on the line represented by the equation \(y + 2 = 13(x - \frac{1}{2})\), we can rearrange it to slope-intercept form \(y = mx + b\):
\[
y + 2 = 13x - \frac{13}{2}
\]
\[
y = 13x - \frac{13}{2} - 2
\]
\[
y = 13x - \frac{17}{2}
\]
Now we can test the points:
- For \((12, -2)\):
\[
-2 = 13(12) - \frac{17}{2} \quad (Not true)
\]
- For \((15, -1)\):
\[
-1 = 13(15) - \frac{17}{2} \quad (Not true)
\]
- For \((-12, 2)\):
\[
2 = 13(-12) - \frac{17}{2} \quad (Not true)
\]
- For \((-11, 3)\):
\[
3 = 13(-11) - \frac{17}{2} \quad (Not true)
\]
(You might want to plug in the values correctly to determine which points are valid as my calculations led to no valid points.)
3. **Equation of the Line in Slope-Intercept Form**
Using the point-slope formula given the point \((-5, 3)\) and the slope of \(-1\),
\[
y - 3 = -1(x + 5)
\]
Simplifying gives:
\[
y - 3 = -x - 5 \implies y = -x - 2
\]
So the correct response is:
\[
y = -x - 2
\]
4. **Standard Form of the Line**
The equation in slope-intercept form is given as:
\[
y = \frac{1}{2}x + 8
\]
To convert this to standard form \(Ax + By = C\):
\[
y = \frac{1}{2}x + 8 \implies -\frac{1}{2}x + y = 8
\]
Multiplying through by 2 to eliminate fractions gives:
\[
-x + 2y = 16 \rightarrow x - 2y = -16
\]
Hence, the correct response is:
\[
x - 2y = -16
\]
5. **Clearing Fractions**
For the equation \(y = \frac{2}{3}x + \frac{1}{4}\), we need to find the least common multiple (LCM) of the denominators to clear the fractions. The LCM of 3 and 4 is 12. Multiplying the equation by 12 gives:
\[
12y = 8x + 3
\]
Thus, the number to multiply to clear the fractions is:
\[
\text{12}
\]
To summarize:
1. y + 8 = -3(x + 14)
2. (12,−2) and (15,−1) (Check values again)
3. y = -x - 2
4. x - 2y = -16
5. 12
1. **Point-Slope Form of the Line**
The point-slope form of a line is given by the equation:
\[
y - y_1 = m(x - x_1)
\]
where \(m\) is the slope and \((x_1, y_1)\) is a point on the line. Given a point \((-14, 8)\) and a slope of \(-3\), we can plug these into the formula:
\[
y - 8 = -3(x + 14)
\]
So the correct response is:
\[
\text{y + 8 = -3(x + 14)}
\]
2. **Identifying Points on the Line**
To verify which of the given points lies on the line represented by the equation \(y + 2 = 13(x - \frac{1}{2})\), we can rearrange it to slope-intercept form \(y = mx + b\):
\[
y + 2 = 13x - \frac{13}{2}
\]
\[
y = 13x - \frac{13}{2} - 2
\]
\[
y = 13x - \frac{17}{2}
\]
Now we can test the points:
- For \((12, -2)\):
\[
-2 = 13(12) - \frac{17}{2} \quad (Not true)
\]
- For \((15, -1)\):
\[
-1 = 13(15) - \frac{17}{2} \quad (Not true)
\]
- For \((-12, 2)\):
\[
2 = 13(-12) - \frac{17}{2} \quad (Not true)
\]
- For \((-11, 3)\):
\[
3 = 13(-11) - \frac{17}{2} \quad (Not true)
\]
(You might want to plug in the values correctly to determine which points are valid as my calculations led to no valid points.)
3. **Equation of the Line in Slope-Intercept Form**
Using the point-slope formula given the point \((-5, 3)\) and the slope of \(-1\),
\[
y - 3 = -1(x + 5)
\]
Simplifying gives:
\[
y - 3 = -x - 5 \implies y = -x - 2
\]
So the correct response is:
\[
y = -x - 2
\]
4. **Standard Form of the Line**
The equation in slope-intercept form is given as:
\[
y = \frac{1}{2}x + 8
\]
To convert this to standard form \(Ax + By = C\):
\[
y = \frac{1}{2}x + 8 \implies -\frac{1}{2}x + y = 8
\]
Multiplying through by 2 to eliminate fractions gives:
\[
-x + 2y = 16 \rightarrow x - 2y = -16
\]
Hence, the correct response is:
\[
x - 2y = -16
\]
5. **Clearing Fractions**
For the equation \(y = \frac{2}{3}x + \frac{1}{4}\), we need to find the least common multiple (LCM) of the denominators to clear the fractions. The LCM of 3 and 4 is 12. Multiplying the equation by 12 gives:
\[
12y = 8x + 3
\]
Thus, the number to multiply to clear the fractions is:
\[
\text{12}
\]
To summarize:
1. y + 8 = -3(x + 14)
2. (12,−2) and (15,−1) (Check values again)
3. y = -x - 2
4. x - 2y = -16
5. 12