Answer:
Equation of the line: y = 1/3x + 4
Explanation:
Identifying the form of -x + 3y = 6 and converting it to slope-intercept form:
- It will be helpful to have -x + 3y = 6 in slope-intercept form to find the equation of the other line.
- First, we need to know its form before we can convert it to slope-intercept form.
-x + 3y = 6 is in the standard form of a line, whose general equation is given by:
Ax + By = C, where:
- A, B, and C are constants.
The general equation of the slope-intercept form is given by:
y = mx + b, where:
- m is the slope,
- and b is the y-intercept.
Thus, we can convert -x + 3y = 6 to slope-intercept form by isolating y on the left-hand side:
(-x + 3y = 6) + x
(3y = x + 6) / 3
y = 1/3x + 2
Thus, the slope of this line is 1/3 and it's y-intercept is 2.
Relationship between the slopes of parallel lines:
- The slopes of parallel are equal to each other.
- This means that the slope of the other line is also 1/3.
Finding the y-intercept of the other line and writing its equation (in slope-intercept form):
Since the other line passes through (3, 5) and its slope is 1/3, we can find its y-intercept (b) by substituting (3, 5) for (x, y) and 1/3 for m in the slope-intercept form:
5 = 1/3(3) + b
(5 = 1 + b) - 1
4 = b
Thus, the y-intercept of the other line is 4.
Therefore, y = 1/3x + 4 is the equation of the line that is parallel to -x + 3y = 6 and passes through the point (3, 5).
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Optional Step:
For the sake of consistency with -x + 3y = 6, you can also convert y = 1/3x + 4 to standard form by isolating 4.
First, we can clear the fraction by multiplying the equation by 3:
3(y = 1/3x + 4)
3y = x + 12
Now, we can subtract x form both sides:
(3y = x + 12) - x
-x + 3y = 12
Therefore, -x + 3y = 12 is the equation of the line (in standard form) that is parallel to -x + 3y = 6 and passes through the point (3, 5).