Write an equation of the line containing the given point and perpendicular to the given line.
(0,4);5x+9y=4
The equation of the line is y=__?
(Simplify your answer. Use integers or fractions for any number in the expression)
The slope of the given line is -5/9
so the slope of the new line must be +9/5
and the new equation will look like
9x- 5y= c
plug in the given point (0,4)
0 - 20 = c
new equation:
9x - 5y = -20
or
y = (9/5)x + 4
To find the equation of a line perpendicular to the given line, we need to determine the slope of the given line and then find the negative reciprocal of that slope.
First, let's rewrite the given line in slope-intercept form (y = mx + b), where m is the slope:
5x + 9y = 4
9y = -5x + 4 (subtract 5x from both sides)
y = (-5/9)x + 4/9 (divide both sides by 9)
Now, we can see that the slope of the given line is -5/9.
The negative reciprocal of -5/9 is 9/5. Therefore, the slope of the line perpendicular to the given line is 9/5.
Next, we use the point-slope formula to find the equation of the line passing through the given point (0,4) with a slope of 9/5:
y - y1 = m(x - x1)
Plugging in the values, we have:
y - 4 = (9/5)(x - 0)
Simplifying, we get:
y - 4 = (9/5)x
5y - 20 = 9x (multiply both sides by 5 to eliminate the fraction)
9x - 5y = -20 (rearrange the terms)
Therefore, the equation of the line perpendicular to the given line and passing through the point (0,4) is 9x - 5y = -20.
To write an equation of the line that is perpendicular to the given line and passes through the point (0,4), we can follow these steps:
Step 1: Determine the slope of the given line.
The given line equation is 5x + 9y = 4. We need to rearrange this equation into slope-intercept form (y = mx + b), where m is the slope. So, let's solve for y:
5x + 9y = 4
9y = -5x + 4
y = (-5/9)x + 4/9
Therefore, the slope of the given line is -5/9.
Step 2: Determine the slope of the line perpendicular to the given line.
The slopes of two perpendicular lines are negative reciprocals of each other. So, the slope of the line perpendicular to the given line will be the negative reciprocal of -5/9. Let's find it:
The negative reciprocal of -5/9 is 9/5.
Step 3: Use the point-slope form to write the equation.
Now that we have the slope of the perpendicular line (9/5) and the given point (0,4), we can use the point-slope form of a line (y - y₁ = m(x - x₁)) to write the equation:
y - 4 = (9/5)(x - 0)
Simplifying:
y - 4 = (9/5)x
Finally, let's rearrange the equation to get it in the form y = __:
y = (9/5)x + 4
Therefore, the equation of the line containing the point (0,4) and perpendicular to the line 5x + 9y = 4 is y = (9/5)x + 4.