Write an equation of the line containing the given point and perpendicular to the given line. Express your answer in the form y=mx+b. (2,4) 3x+y=6 The equation of line is y=, and explain how you have gotten the answer.
1. First you need to find the slope of the original line you are given which is 3x + y = 6. The slope is -A/B so it is -3/1 or just -3, m = -3.
2. The slope of a line perpendicular is the negative reciprocal so the slope of the line perpendicular to the given line is 1/3.
3. Use point slope form with the given point to find the equation of the line first before simplifying it to y=mx+b.
Point slope form is y - y1 = m(x-x1) now substitute the point
y-4=1/3(x-2)
4. Now distribute the 1/3: y-4=1/3x - 2/3
Then add 4 to the other side, make 4 into 12/3 to give it a command denominator making it easier to add.
y = 1/3x + 10/3.
Thank You so much Elle
To find the equation of a line perpendicular to a given line, we need to determine the slope of the given line first.
Given line: 3x + y = 6
To express this equation in slope-intercept form (y = mx + b), we need to solve for y by subtracting 3x from both sides:
y = -3x + 6
Now, we can determine the slope of the given line by observing the coefficient of x, which is -3. The slope of a line perpendicular to this line will be the negative reciprocal of -3, which is 1/3.
Now that we have the slope of the perpendicular line, we can use the given point (2,4) to find the equation.
Using the point-slope form of a line, which states that y - y1 = m(x - x1), where (x1, y1) is a given point on the line, we substitute the values m = 1/3, x1 = 2, and y1 = 4:
y - 4 = (1/3)(x - 2)
To express this equation in slope-intercept form, we can distribute 1/3 to (x - 2):
y - 4 = (1/3)x - 2/3
Now, isolate y by adding 4 to both sides:
y = (1/3)x - 2/3 + 4
Simplify:
y = (1/3)x + 10/3
Therefore, the equation of the line containing the point (2,4) and perpendicular to the line 3x + y = 6 is y = (1/3)x + 10/3.
To find the equation of a line that is perpendicular to another line, we need to determine the slope of the given line and then find the negative reciprocal of that slope. Let's start with the given line: 3x + y = 6.
To convert this equation into the slope-intercept form (y = mx + b), we need to isolate y.
First, subtract 3x from both sides of the equation:
y = -3x + 6.
Now we can see that the slope of the given line is -3, since the coefficient of x is the slope (m).
To find the slope of the line perpendicular to this one, we take the negative reciprocal of -3. The negative reciprocal of a number is found by flipping its sign and then taking the reciprocal (1 divided by that number).
So, the negative reciprocal of -3 is (-1/3).
Now we have the slope of the line perpendicular to the given line, which is -1/3.
We also have a point that the perpendicular line passes through, which is (2, 4).
Using the point-slope form of the equation of a line, we can write the equation as:
y - y1 = m(x - x1),
where (x1, y1) is the given point and m is the slope.
Plugging in the values, we have:
y - 4 = (-1/3)(x - 2).
To simplify, we can distribute the (-1/3) to (x - 2):
y - 4 = (-1/3)x + (2/3).
Finally, add 4 to both sides of the equation to isolate y:
y = (-1/3)x + (2/3) + 4.
Combining the constants on the right side gives us:
y = (-1/3)x + 14/3.
Therefore, the equation of the line containing the point (2, 4) and perpendicular to the line 3x + y = 6 is y = (-1/3)x + 14/3.
Explanation Recap:
- Converted the given line equation to slope-intercept form.
- Identified the slope of the given line as -3.
- Found the negative reciprocal of -3, which is -1/3, as the slope of the perpendicular line.
- Utilized the point-slope form using the given point (2, 4) and the perpendicular slope.
- Simplified the equation to y = (-1/3)x + 14/3.