Find the equation of the line
With a slope of 2/3 that goes through the point (4, 5)
With a slope of (-1/3) that goes through the point (2, -4)
With a slope of 3 that goes through the point (-5, -1)
I would do the first one this way.
The equation for a straight line is
y = mx + b where m is the slope and b is the y intercept. If you want the line to go through (4,5) and slope = m = 2/3, then
set y = 5, m = 2/3, and x = 4
5=(2/3)*4 + b
Solve for b.
Multiply both sides by 3 to clear the fraction.
15 = 8 + 3b
3b = 7 and b = 7/3 so th equation is
y = (2/3)x + (7/3) or to get rid of the fractions, it would be
3y = 2x + 7
CHECK:
plug in 4 for x and 5 for y
3*5=2*4 + 7
15 = 8 + 7
15 = 15.
The others are done the same way.
Check my work.
To find the equation of a line, we can use the slope-intercept form of a linear equation, which is given by:
y = mx + b
where m is the slope of the line, and b is the y-intercept.
Let's go through each question and find the equation of the line.
1. With a slope of 2/3 that goes through the point (4, 5):
First, substitute the slope (m) and the coordinates of the point (x, y) into the slope-intercept form:
5 = (2/3)*4 + b
Next, simplify:
5 = 8/3 + b
To solve for b, subtract 8/3 from both sides:
5 - 8/3 = b
To get a common denominator, convert 5 to 15/3:
15/3 - 8/3 = b
Now subtract:
7/3 = b
The equation of the line is therefore:
y = (2/3)x + 7/3
2. With a slope of -1/3 that goes through the point (2, -4):
Using the slope-intercept form, we have:
-4 = (-1/3)*2 + b
Next, simplify:
-4 = -2/3 + b
To solve for b, add 2/3 to both sides:
-4 + 2/3 = b
To get a common denominator, convert -4 to -12/3:
-12/3 + 2/3 = b
Now add:
-10/3 = b
The equation of the line is therefore:
y = (-1/3)x - 10/3
3. With a slope of 3 that goes through the point (-5, -1):
Using the slope-intercept form, we have:
-1 = 3*(-5) + b
Next, simplify:
-1 = -15 + b
To solve for b, add 15 to both sides:
-1 + 15 = b
-1 + 15 simplifies to 14, so we have:
14 = b
The equation of the line is therefore:
y = 3x + 14