express in y=mx + b form the equation of the line perpendicular to the line 5x-3y=8 and passes through the piont (-5,1)
To express the equation of a line in the form y = mx + b, where m represents the slope and b represents the y-intercept, we need to find the slope and y-intercept values.
Given the equation of the line 5x - 3y = 8, we need to first rewrite it in the slope-intercept form.
Step 1: Rewrite the equation in slope-intercept form.
Start by isolating y on one side of the equation:
5x - 3y = 8
-3y = -5x + 8
Divide both sides by -3 to obtain:
y = (5/3)x - 8/3
Now we can determine the slope m of the original line. By comparing y = (5/3)x - 8/3 to y = mx + b, we see that m = 5/3.
To find the slope of the line perpendicular to the original line, we can use the fact that perpendicular lines have slopes that are negative reciprocals of each other. The negative reciprocal of 5/3 can be obtained by flipping the fraction and changing its sign, so the slope of the perpendicular line is -3/5.
Now we have the slope of the perpendicular line and a point it must pass through, (-5, 1). We can use the point-slope formula to find the equation.
Step 2: Apply the point-slope formula.
The point-slope formula is given as:
y - y₁ = m(x - x₁)
Substitute the values we know:
y - 1 = (-3/5)(x - (-5))
Simplify:
y - 1 = (-3/5)(x + 5)
Distribute -3/5 across (x + 5):
y - 1 = (-3/5)x - 3
Add 1 to both sides to isolate y:
y = (-3/5)x - 3 + 1
y = (-3/5)x - 2
Therefore, the equation of the line perpendicular to 5x - 3y = 8 and passing through the point (-5, 1) is y = (-3/5)x - 2.