which equation is perpendicular to 2y = 8x+7?
The Correct Answer and Explanation is:
To find an equation that is perpendicular to 2y=8x+72y = 8x + 72y=8x+7, we first need to rewrite this equation in slope-intercept form, y=mx+by = mx + by=mx+b, where mmm represents the slope of the line.
Step 1: Rewrite the given equation in slope-intercept form
Starting with the equation:2y=8x+72y = 8x + 72y=8x+7
Divide both sides by 2 to solve for yyy:y=4x+72y = 4x + \frac{7}{2}y=4x+27
Now we have the slope of the given line, which is m=4m = 4m=4.
Step 2: Determine the slope of the perpendicular line
The slopes of two perpendicular lines are negative reciprocals of each other. This means that if one line has slope m1m_1m1, the other line’s slope m2m_2m2 is given by:m2=−1m1m_2 = -\frac{1}{m_1}m2=−m11
For our line, the slope m1=4m_1 = 4m1=4. Therefore, the slope of the line perpendicular to this one will be:m2=−14m_2 = -\frac{1}{4}m2=−41
Step 3: Write the equation of the perpendicular line
Now that we know the slope of the perpendicular line is m2=−14m_2 = -\frac{1}{4}m2=−41, we can write its equation in slope-intercept form:y=−14x+by = -\frac{1}{4}x + by=−41x+b
Here, bbb is the y-intercept, which can be any value depending on where the perpendicular line intersects the y-axis. So, any equation of the form:y=−14x+by = -\frac{1}{4}x + by=−41x+b
where bbb is a constant, will be perpendicular to the original equation.
In conclusion, the equation perpendicular to 2y=8x+72y = 8x + 72y=8x+7 is any equation where the slope is −14-\frac{1}{4}−41, such as y=−14x+by = -\frac{1}{4}x + by=−41x+b.
