what is an equation of the line containing the given point and prallel to the given line; (5,-2);6x-7y=4
First, put the equation in slope-intercept form.
6x-7y=4
-7y=-6x+4
-y=-6/7x+4/7
y=6/7x-4/7
So the slope of the parallel line will also be 6/7.
Now use the point slope formula to find the
equation of the line.
y-y1=m(x-x1)
y-(-2)=6/7(x-5)
y+2=6/7(x-5)
y+2=6/7x-30/7
y=6/7x-44/7
That is the equation of the line.
To find the equation of a line parallel to a given line through a given point, we can follow these steps:
Step 1: Determine the slope of the given line.
In order to find the slope-intercept form of a line, we need to rewrite the given equation in the form: y = mx + b, where m represents the slope.
So, let's rewrite the given equation:
6x - 7y = 4
First, subtract 6x from both sides to isolate the term with y:
-7y = -6x + 4
Next, divide all the terms by -7 to solve for y:
y = (6/7)x - 4/7
The slope of the given line is 6/7.
Step 2: Use the given point and the slope to find the equation of the line.
Now that we have the slope of the given line (6/7), we can use the point (5,-2) to find the equation of the line parallel to it.
We'll use the point-slope form of a linear equation: y - y₁ = m(x - x₁), where (x₁, y₁) is the given point and m is the slope.
Let's substitute the values into the equation:
y - (-2) = (6/7)(x - 5)
Next, simplify and solve for y:
y + 2 = (6/7)(x - 5)
To have the equation in slope-intercept form (y = mx + b), we need to isolate y.
Distribute (6/7) to (x - 5), and simplify:
y + 2 = (6/7)x - 30/7
Subtract 2 from both sides:
y = (6/7)x - 30/7 - 14/7
y = (6/7)x - 44/7
Therefore, the equation of the line parallel to the given line (6x - 7y = 4), and passing through the point (5,-2), is:
y = (6/7)x - 44/7.
To find an equation of a line that is parallel to a given line, we need to know that parallel lines have the same slope. Therefore, we need to determine the slope of the given line.
The equation of the given line is 6x - 7y = 4. To find the slope, we can rearrange the equation into slope-intercept form, y = mx + b, where m is the slope.
Rearranging the equation, we get:
-7y = -6x + 4
Divide both sides by -7 to isolate y:
y = (6/7)x - 4/7
From this equation, we can see that the slope of the given line is 6/7.
Now that we know the slope of the given line, we can find the equation of the line parallel to it that passes through the point (5, -2).
Using the point-slope form of a linear equation, y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line and m is the slope, we can substitute the values into the equation.
Given:
Point (x₁, y₁) = (5, -2)
Slope m = 6/7
Plugging these values into the equation:
y - (-2) = (6/7)(x - 5)
Simplifying:
y + 2 = (6/7)(x - 5)
Now, we can distribute the 6/7 to the terms within the parentheses:
y + 2 = (6/7)x - (6/7)(5)
Simplifying further:
y + 2 = (6/7)x - 30/7
To get the equation in standard form, we want to eliminate any fractions by multiplying through by the common denominator, which in this case is 7:
7(y + 2) = 7(6/7)x - 7(30/7)
Simplifying again:
7y + 14 = 6x - 30
Rearranging the equation to match the standard form:
6x - 7y = 44
Therefore, an equation of the line containing the point (5, -2) and parallel to the line 6x - 7y = 4 is 6x - 7y = 44.