Question
What is the point-slope form of the equation of the line that has a slope of 2 and passes through the point (−5,4)
?(1 point)./
y−
=2
(x−
)
?(1 point)./
y−
=2
(x−
)
Answers
GPT-4o mini
The point-slope form of a line is given by the formula:
\[
y - y_1 = m(x - x_1)
\]
where \( m \) is the slope and \((x_1, y_1)\) is a point on the line.
Given that the slope \( m = 2 \) and the point is \((x_1, y_1) = (-5, 4)\), we can substitute these values into the formula:
\[
y - 4 = 2(x - (-5))
\]
This simplifies to:
\[
y - 4 = 2(x + 5)
\]
So, the point-slope form of the equation of the line is:
\[
y - 4 = 2(x + 5)
\]
In your requested format:
\[
y - 4 = 2(x - (-5))
\]
Thus, the answer is:
- \( y - 4 \)
- \( 2 \)
- \( -5 \)
\[
y - y_1 = m(x - x_1)
\]
where \( m \) is the slope and \((x_1, y_1)\) is a point on the line.
Given that the slope \( m = 2 \) and the point is \((x_1, y_1) = (-5, 4)\), we can substitute these values into the formula:
\[
y - 4 = 2(x - (-5))
\]
This simplifies to:
\[
y - 4 = 2(x + 5)
\]
So, the point-slope form of the equation of the line is:
\[
y - 4 = 2(x + 5)
\]
In your requested format:
\[
y - 4 = 2(x - (-5))
\]
Thus, the answer is:
- \( y - 4 \)
- \( 2 \)
- \( -5 \)
A line passes through the point (6,13)
and has a slope of −2
. What is the equation of the line in slope-intercept form?(1 point)
y=
x+
and has a slope of −2
. What is the equation of the line in slope-intercept form?(1 point)
y=
x+
GPT-4o mini
To find the equation of a line in slope-intercept form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept, we can start by using the point-slope form of a line:
The point-slope form is given by:
\[
y - y_1 = m(x - x_1)
\]
Where:
- \( m = -2 \) (the slope)
- \((x_1, y_1) = (6, 13)\) (the point)
Substituting these values into the point-slope formula:
\[
y - 13 = -2(x - 6)
\]
Now, we'll simplify this equation:
1. Distribute \(-2\):
\[
y - 13 = -2x + 12
\]
2. Add 13 to both sides:
\[
y = -2x + 12 + 13
\]
\[
y = -2x + 25
\]
Thus, in slope-intercept form, the equation of the line is:
\[
y = -2x + 25
\]
So, in your requested format:
- \( y = \)
- \( -2 \)
- \( 25 \)
The point-slope form is given by:
\[
y - y_1 = m(x - x_1)
\]
Where:
- \( m = -2 \) (the slope)
- \((x_1, y_1) = (6, 13)\) (the point)
Substituting these values into the point-slope formula:
\[
y - 13 = -2(x - 6)
\]
Now, we'll simplify this equation:
1. Distribute \(-2\):
\[
y - 13 = -2x + 12
\]
2. Add 13 to both sides:
\[
y = -2x + 12 + 13
\]
\[
y = -2x + 25
\]
Thus, in slope-intercept form, the equation of the line is:
\[
y = -2x + 25
\]
So, in your requested format:
- \( y = \)
- \( -2 \)
- \( 25 \)
Nico is trying to write an equation in slope-intercept form for a line that passes through the point (1,2)
and has a slope of −12
. What is the y-intercept of this line, rounded to the nearest tenth? (1 point)
and has a slope of −12
. What is the y-intercept of this line, rounded to the nearest tenth? (1 point)
GPT-4o mini
To write the equation of a line in slope-intercept form \( y = mx + b \), we need to know the slope \( m \) and the y-intercept \( b \).
Given that:
- The slope \( m = -12 \)
- The line passes through the point \( (1, 2) \), which means \( x_1 = 1 \) and \( y_1 = 2 \)
We can substitute these values into the slope-intercept form of the equation to find the y-intercept \( b \).
1. Start with the slope-intercept form:
\[
y = mx + b
\]
2. Substitute the point \((1, 2)\) and the slope \( -12\) into the equation:
\[
2 = -12(1) + b
\]
3. Simplify:
\[
2 = -12 + b
\]
4. Add 12 to both sides to isolate \( b \):
\[
b = 2 + 12
\]
\[
b = 14
\]
Thus, the y-intercept of the line is \( 14.0 \).
In summary, rounded to the nearest tenth, the y-intercept \( b \) is:
\[
\boxed{14.0}
\]
Given that:
- The slope \( m = -12 \)
- The line passes through the point \( (1, 2) \), which means \( x_1 = 1 \) and \( y_1 = 2 \)
We can substitute these values into the slope-intercept form of the equation to find the y-intercept \( b \).
1. Start with the slope-intercept form:
\[
y = mx + b
\]
2. Substitute the point \((1, 2)\) and the slope \( -12\) into the equation:
\[
2 = -12(1) + b
\]
3. Simplify:
\[
2 = -12 + b
\]
4. Add 12 to both sides to isolate \( b \):
\[
b = 2 + 12
\]
\[
b = 14
\]
Thus, the y-intercept of the line is \( 14.0 \).
In summary, rounded to the nearest tenth, the y-intercept \( b \) is:
\[
\boxed{14.0}
\]
Nico is trying to write an equation in slope-intercept form for a line that passes through the point (1,2)
and has a slope of −1/2
. What is the y-intercept of this line, rounded to the nearest tenth? (1 point)
and has a slope of −1/2
. What is the y-intercept of this line, rounded to the nearest tenth? (1 point)
GPT-4o mini
To find the equation of the line in slope-intercept form \( y = mx + b \) and determine the y-intercept \( b \), we will use the given slope and the point the line passes through.
Given:
- Slope \( m = -\frac{1}{2} \)
- Point \( (1, 2) \), which means \( x_1 = 1 \) and \( y_1 = 2 \)
We can substitute these values into the slope-intercept form:
1. Start with the equation:
\[
y = mx + b
\]
2. Substitute the point \((1, 2)\) and the slope \(-\frac{1}{2}\):
\[
2 = -\frac{1}{2}(1) + b
\]
3. Simplify:
\[
2 = -\frac{1}{2} + b
\]
4. Solve for \( b \) by adding \(\frac{1}{2}\) to both sides:
\[
b = 2 + \frac{1}{2}
\]
5. Convert \(2\) to a fraction with a common denominator of 2:
\[
b = \frac{4}{2} + \frac{1}{2} = \frac{5}{2}
\]
6. Convert \(\frac{5}{2}\) to a decimal:
\[
\frac{5}{2} = 2.5
\]
Thus, the y-intercept of the line is \( 2.5 \).
Rounded to the nearest tenth, the answer is:
\[
\boxed{2.5}
\]
Given:
- Slope \( m = -\frac{1}{2} \)
- Point \( (1, 2) \), which means \( x_1 = 1 \) and \( y_1 = 2 \)
We can substitute these values into the slope-intercept form:
1. Start with the equation:
\[
y = mx + b
\]
2. Substitute the point \((1, 2)\) and the slope \(-\frac{1}{2}\):
\[
2 = -\frac{1}{2}(1) + b
\]
3. Simplify:
\[
2 = -\frac{1}{2} + b
\]
4. Solve for \( b \) by adding \(\frac{1}{2}\) to both sides:
\[
b = 2 + \frac{1}{2}
\]
5. Convert \(2\) to a fraction with a common denominator of 2:
\[
b = \frac{4}{2} + \frac{1}{2} = \frac{5}{2}
\]
6. Convert \(\frac{5}{2}\) to a decimal:
\[
\frac{5}{2} = 2.5
\]
Thus, the y-intercept of the line is \( 2.5 \).
Rounded to the nearest tenth, the answer is:
\[
\boxed{2.5}
\]
A line with a slope of m=−1/2
passes through the point (4,6)
. What is the equation of the line in standard form?(1 point)
passes through the point (4,6)
. What is the equation of the line in standard form?(1 point)
GPT-4o mini
To determine the equation of a line in standard form \(Ax + By = C\) given the slope \(m = -\frac{1}{2}\) and the point \((4, 6)\), we can follow these steps:
1. **Start with the point-slope form of the line**:
\[
y - y_1 = m(x - x_1)
\]
Substituting in the given slope and point \((x_1, y_1) = (4, 6)\):
\[
y - 6 = -\frac{1}{2}(x - 4)
\]
2. **Simplify the equation**:
Distributing \(-\frac{1}{2}\):
\[
y - 6 = -\frac{1}{2}x + 2
\]
Now, add 6 to both sides:
\[
y = -\frac{1}{2}x + 8
\]
3. **Convert to standard form**:
To get this into standard form \(Ax + By = C\), we can eliminate the fraction by multiplying through by 2:
\[
2y = -x + 16
\]
Rearranging gives:
\[
x + 2y = 16
\]
Now, \(A = 1\), \(B = 2\), and \(C = 16\), which satisfies the conditions of standard form where \(A\) is positive.
The final equation in standard form is:
\[
\boxed{x + 2y = 16}
\]
1. **Start with the point-slope form of the line**:
\[
y - y_1 = m(x - x_1)
\]
Substituting in the given slope and point \((x_1, y_1) = (4, 6)\):
\[
y - 6 = -\frac{1}{2}(x - 4)
\]
2. **Simplify the equation**:
Distributing \(-\frac{1}{2}\):
\[
y - 6 = -\frac{1}{2}x + 2
\]
Now, add 6 to both sides:
\[
y = -\frac{1}{2}x + 8
\]
3. **Convert to standard form**:
To get this into standard form \(Ax + By = C\), we can eliminate the fraction by multiplying through by 2:
\[
2y = -x + 16
\]
Rearranging gives:
\[
x + 2y = 16
\]
Now, \(A = 1\), \(B = 2\), and \(C = 16\), which satisfies the conditions of standard form where \(A\) is positive.
The final equation in standard form is:
\[
\boxed{x + 2y = 16}
\]
The equation of a line is given as 1/3x+1/6y=4
. If Rashonda wants to clear the fractions, by what number should she multiply the equation?(1 point)
. If Rashonda wants to clear the fractions, by what number should she multiply the equation?(1 point)
GPT-4o mini
To eliminate the fractions from the equation
\[
\frac{1}{3}x + \frac{1}{6}y = 4,
\]
we need to find a common multiple of the denominators \(3\) and \(6\). The least common multiple (LCM) of \(3\) and \(6\) is \(6\).
By multiplying the entire equation by \(6\), we can eliminate the fractions:
\[
6 \left(\frac{1}{3}x\right) + 6 \left(\frac{1}{6}y\right) = 6 \times 4.
\]
Doing this gives:
\[
2x + y = 24.
\]
Thus, Rashonda should multiply the equation by
\[
\boxed{6}
\] to clear the fractions.
\[
\frac{1}{3}x + \frac{1}{6}y = 4,
\]
we need to find a common multiple of the denominators \(3\) and \(6\). The least common multiple (LCM) of \(3\) and \(6\) is \(6\).
By multiplying the entire equation by \(6\), we can eliminate the fractions:
\[
6 \left(\frac{1}{3}x\right) + 6 \left(\frac{1}{6}y\right) = 6 \times 4.
\]
Doing this gives:
\[
2x + y = 24.
\]
Thus, Rashonda should multiply the equation by
\[
\boxed{6}
\] to clear the fractions.