write an equation of the line containing the given point and parallel to the given line (9,-9); 3x-5y=8
The most useful form of straight-line equations is the "slope-intercept" form:
y = mx + b
m is the slope and "b" gives the
y-intercept.
3x-5y=8
-5y=8-3x Divide with -5
y=(-8/5)+(3x/5)
y=(3/5)x-8/5
The other format for straight-line equations is called the "point-slope" form. For this one, they give you a point (x1, y1) and a slope m, and have you plug it into this formula:
y-y1= m(x – x1)
Parallel lines have same slope.
In this case:
m=3/5
x1=9
y1= -9
y-y1= m(x – x1)
y-(-9)=(3/5)*(x-9)
y+9=(3/5)x-(3*9/5)
y=(3/5)x-(27/5)-9
y=(3/5)x-(27/5)-(45/5)
y=(3/5)x-72/5
OR:
y-(3/5)x= -72/5 Multiply with -5
-5y+3x=72
3x-5y=72
y=(1/4)x-6
To find the equation of a line parallel to the given line, we need to determine the slope of the given line first, as parallel lines have the same slope.
The given line has the equation 3x - 5y = 8. To determine its slope, we need to rearrange the equation in slope-intercept form, which is y = mx + b (m represents the slope, and b is the y-intercept).
Rearranging the equation, we get:
-5y = -3x + 8
Divide both sides by -5 to isolate y:
y = (3/5)x - 8/5
From the equation, we can see that the slope of the given line is 3/5. Therefore, any line parallel to this line will also have a slope of 3/5.
Now that we have the slope, we can use it along with the given point (9,-9) to determine the equation of the line using the point-slope form:
y - y1 = m(x - x1)
Substituting the values, we get:
y - (-9) = (3/5)(x - 9)
Simplifying further:
y + 9 = (3/5)(x - 9)
Expanding and rearranging:
y + 9 = (3/5)x - (3/5)(9)
y + 9 = (3/5)x - (27/5)
y = (3/5)x - (27/5) - 9
y = (3/5)x - 27/5 - 45/5
y = (3/5)x - 72/5
Therefore, the equation of the line parallel to 3x - 5y = 8 and containing the point (9,-9) is y = (3/5)x - 72/5.