Question
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
Answers
Answered by
GPT-4o mini
To write the equation of the line in standard form, we start with the point-slope form of the line equation:
\[
y - y_1 = m(x - x_1)
\]
Where \((x_1, y_1)\) is the point on the line and \(m\) is the slope. For your line:
- The point \((x_1, y_1)\) is \((2, 9)\)
- The slope \(m\) is \(\frac{1}{2}\)
Substituting these values into the point-slope form:
\[
y - 9 = \frac{1}{2}(x - 2)
\]
Now, we simplify this equation:
\[
y - 9 = \frac{1}{2}x - 1
\]
Next, we'll add 9 to both sides:
\[
y = \frac{1}{2}x + 8
\]
Now, we'll convert this equation into standard form \(Ax + By = C\). First, multiply every term by 2 to eliminate the fraction:
\[
2y = x + 16
\]
Next, rearrange the equation to the standard form \(Ax + By = C\):
\[
-x + 2y = 16
\]
To express this with a positive \(A\), we can multiply the entire equation by -1:
\[
x - 2y = -16
\]
Thus, the equation of the line in standard form is:
\[
x - 2y = -16
\]
\[
y - y_1 = m(x - x_1)
\]
Where \((x_1, y_1)\) is the point on the line and \(m\) is the slope. For your line:
- The point \((x_1, y_1)\) is \((2, 9)\)
- The slope \(m\) is \(\frac{1}{2}\)
Substituting these values into the point-slope form:
\[
y - 9 = \frac{1}{2}(x - 2)
\]
Now, we simplify this equation:
\[
y - 9 = \frac{1}{2}x - 1
\]
Next, we'll add 9 to both sides:
\[
y = \frac{1}{2}x + 8
\]
Now, we'll convert this equation into standard form \(Ax + By = C\). First, multiply every term by 2 to eliminate the fraction:
\[
2y = x + 16
\]
Next, rearrange the equation to the standard form \(Ax + By = C\):
\[
-x + 2y = 16
\]
To express this with a positive \(A\), we can multiply the entire equation by -1:
\[
x - 2y = -16
\]
Thus, the equation of the line in standard form is:
\[
x - 2y = -16
\]
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