We are given the point (-2,4) and the line y=2x+9. We want the equation of the line that passes through the given point and that is perpendicular to the given line.
To do so, we will use the following equation of a line
[tex]y\text{ -a = m\lparen x -b\rparen}[/tex]in this equation, m is the slope of the line and (a,b) is a point in the line. In our case, we are given that (-2,4) is in the line. That is, a=-2 and b=4. So our equation becomes
[tex]y\text{ -4=m\lparen x -\lparen-2\rparen\rparen}[/tex]or equivalently
[tex]y\text{ -4}=m(x\text{ +2\rparen}[/tex]now, we only need to find the value of m. To do so, we use the given line and the fact that the product of the slopes of perpendicular lines is -1.
The given line (2x+9) has a slope of 2. So, we have the following equation
[tex]m\cdot2=\text{ -1}[/tex]so if we divide both sides by 2, we get that
[tex]m=\text{ -}\frac{1}{2}[/tex]So the equation we are looking for becomes
[tex]y\text{ -4 }=\text{ -}\frac{1}{2}(x\text{ +2\rparen}[/tex]We want this equation in the slope intercept form. So we operate to find y in this equation. So first, we distribute on the right hand side. We get
[tex]y\text{ -4}=\text{ -}\frac{1}{2}x\text{ -}\frac{2}{2}=\text{ -}\frac{1}{2}x\text{ -1}[/tex]now we add 4 on both sides, so we get
[tex]y=\text{ -}\frac{1}{2}x\text{ -1+4= -}\frac{1}{2}x+3[/tex]we can check that if x= -2 we get
[tex]y=\text{ -}\frac{1}{2}(\text{ -2\rparen+3=1+3=4}[/tex]which confirms that the point (-2,4) is on the line
