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
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
Answers
Answered by
GPT-4o mini
Let's go through each of your questions one by one.
1. **Point-Slope Form:**
The point-slope form of a line is given by the equation:
\( y - y_1 = m(x - x_1) \)
where \( (x_1, y_1) \) is a point on the line and \( m \) is the slope.
Given the point (−14, 8) and a slope of −3, we can substitute these values into the formula:
\( y - 8 = -3(x + 14) \)
Therefore, the correct response is:
**y + 8 = −3(x + 14)**
2. **Points on the Line:**
To determine which set of points lies on the line given by \( y + 2 = 13(x - \frac{1}{2}) \), we can check each set by plugging the \( x \)-values into the equation and seeing if we get the corresponding \( y \)-values.
Let's test the points:
- For \( (12, −2) \):
Substitute \( x = 12 \):
\( y + 2 = 13(12 - \frac{1}{2}) \)
\( y + 2 = 13(11.5) \)
\( y + 2 = 149.5 \)
\( y = 147.5 \) (not equal to -2)
- For \( (15, −1) \):
Substitute \( x = 15 \):
\( y + 2 = 13(15 - \frac{1}{2}) \)
\( y + 2 = 13(14.5) \)
\( y + 2 = 188.5 \)
\( y = 186.5 \) (not equal to -1)
- And so on for the other sets.
You should test each option in a similar way — I suggest using a calculator or checking systematically, but it's clearer if you manually work through options to find which points satisfy the equation.
3. **Slope-Intercept Form:**
From the point-slope form, the equation with point (-5, 3) and slope -1 can be written as:
\( y - 3 = -1(x + 5) \)
\( y - 3 = -x - 5 \)
\( y = -x - 2 \)
Thus, the correct response is:
**y = −x − 2**
4. **Standard Form:**
The slope-intercept form \( y = \frac{1}{2}x + 8 \) can be transformed into standard form (Ax + By = C) by eliminating the fraction. Multiply through by 2:
\( 2y = x + 16 \)
Rearranging gives us:
\( -x + 2y = 16 \) or in standard form \( x - 2y = -16 \) (same line, different representation).
The correct answer is:
**x - 2y = -16**
5. **Clearing Fractions:**
The equation \( y = \frac{2}{3}x + \frac{1}{4} \) needs to have both fractions multiplied out. The least common multiple of the denominators (3 and 4) is 12.
Multiplying through by 12 clears the fractions.
Therefore, the correct response is:
**12**
You can refer back to each question for confirmation on how the answers fit together with the respective forms and transformations.
1. **Point-Slope Form:**
The point-slope form of a line is given by the equation:
\( y - y_1 = m(x - x_1) \)
where \( (x_1, y_1) \) is a point on the line and \( m \) is the slope.
Given the point (−14, 8) and a slope of −3, we can substitute these values into the formula:
\( y - 8 = -3(x + 14) \)
Therefore, the correct response is:
**y + 8 = −3(x + 14)**
2. **Points on the Line:**
To determine which set of points lies on the line given by \( y + 2 = 13(x - \frac{1}{2}) \), we can check each set by plugging the \( x \)-values into the equation and seeing if we get the corresponding \( y \)-values.
Let's test the points:
- For \( (12, −2) \):
Substitute \( x = 12 \):
\( y + 2 = 13(12 - \frac{1}{2}) \)
\( y + 2 = 13(11.5) \)
\( y + 2 = 149.5 \)
\( y = 147.5 \) (not equal to -2)
- For \( (15, −1) \):
Substitute \( x = 15 \):
\( y + 2 = 13(15 - \frac{1}{2}) \)
\( y + 2 = 13(14.5) \)
\( y + 2 = 188.5 \)
\( y = 186.5 \) (not equal to -1)
- And so on for the other sets.
You should test each option in a similar way — I suggest using a calculator or checking systematically, but it's clearer if you manually work through options to find which points satisfy the equation.
3. **Slope-Intercept Form:**
From the point-slope form, the equation with point (-5, 3) and slope -1 can be written as:
\( y - 3 = -1(x + 5) \)
\( y - 3 = -x - 5 \)
\( y = -x - 2 \)
Thus, the correct response is:
**y = −x − 2**
4. **Standard Form:**
The slope-intercept form \( y = \frac{1}{2}x + 8 \) can be transformed into standard form (Ax + By = C) by eliminating the fraction. Multiply through by 2:
\( 2y = x + 16 \)
Rearranging gives us:
\( -x + 2y = 16 \) or in standard form \( x - 2y = -16 \) (same line, different representation).
The correct answer is:
**x - 2y = -16**
5. **Clearing Fractions:**
The equation \( y = \frac{2}{3}x + \frac{1}{4} \) needs to have both fractions multiplied out. The least common multiple of the denominators (3 and 4) is 12.
Multiplying through by 12 clears the fractions.
Therefore, the correct response is:
**12**
You can refer back to each question for confirmation on how the answers fit together with the respective forms and transformations.