Answer:
![y=-\frac{1}{2}x+2](https://tex.z-dn.net/?f=y%3D-%5Cfrac%7B1%7D%7B2%7Dx%2B2)
Step-by-step explanation:
So we want to find the equation of a line perpendicular to y=2x+5 and passes through the point (-6,5).
First, let's determine the slope of our equation. Remember that the slopes of perpendicular lines are negative reciprocals. In other words:
![m_1\cdot m_2=-1](https://tex.z-dn.net/?f=m_1%5Ccdot%20m_2%3D-1)
To find our slope, let's substitute 2 (the slope of y=2x+5) for m₁ and solve for m₂. So:
![2\cdot m_2=-1](https://tex.z-dn.net/?f=2%5Ccdot%20m_2%3D-1)
Divide both sides by 2:
![m_2=-\frac{1}{2}](https://tex.z-dn.net/?f=m_2%3D-%5Cfrac%7B1%7D%7B2%7D)
Therefore, the slope of our new line is -1/2.
Now, we can use the point-slope form to find the equation of our line. The point-slope form is:
![y-y_1=m(x-x_1)](https://tex.z-dn.net/?f=y-y_1%3Dm%28x-x_1%29)
Where m is the slope and (x₁, y₂) is a point.
So, let's substitute -1/2 for m and (-6,5) for (x₁, y₂), respectively. So:
![y-5=-\frac{1}{2}(x-(-6))](https://tex.z-dn.net/?f=y-5%3D-%5Cfrac%7B1%7D%7B2%7D%28x-%28-6%29%29)
Simplify:
![y-5=-\frac{1}{2}(x+6)](https://tex.z-dn.net/?f=y-5%3D-%5Cfrac%7B1%7D%7B2%7D%28x%2B6%29)
Distribute:
![y-5=-\frac{1}{2}x-3](https://tex.z-dn.net/?f=y-5%3D-%5Cfrac%7B1%7D%7B2%7Dx-3)
Add 5 to both sides:
![y=-\frac{1}{2}x+2](https://tex.z-dn.net/?f=y%3D-%5Cfrac%7B1%7D%7B2%7Dx%2B2)
So, our equation is:
![y=-\frac{1}{2}x+2](https://tex.z-dn.net/?f=y%3D-%5Cfrac%7B1%7D%7B2%7Dx%2B2)