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Write an equation of the line tht passes through the given point and is perpendicular to the graph of the given equation?

Write an equation of the line that passes through the given point and is perpendicular to the graph of the given equation?
1. (-9, 3); 3x + y = 5
2. (-8, 3); y +4 = -2/3(x-2)
3. (3, -6); x + y = -4

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4 Answers

Let's take a look at the frist problem. 2 and 3 can be solved the same way.

First of all you have to rewrite your equation 3x + y = 5 (this is a standard form) in slope-intercept form by subtracting 3x from both sides of the equation. We have y = -3x +5. That means the sloppe of the line is

m=-3 (if we use y = mx+b form of the linear equation). The line which is perpendicular to that will have the slope equal to the reciprocal of (-3) taken with negative sign. Thus, the  new slope equals 1/3. Or your equation for the perpendicular line should be like

                                                           y = (1/3)x +b

In order to find "b" use the point given in the problem. If x =-9 y =3. Thus, we have

                                   3 = (1/3)(-9) +b = -3 +b        or b = 3+3 =6

The equation for  a perpendicular line becomes

                                            y = (1/3)x +6

Repeat the same steps for problems 2 and 3.

1. The equation can be rewritten y = 5 - 3x, which is in slope-intercept form. This makes it obvious that the slope of the line is -3. To find the slope of a line perpendicular to it, you take the negative reciprocal of the known slope. -(-3)^(-1) = 1/3 which is the slope we desire. We can write a new equation in point slope form: y - y0 = m(x-x0) where (x0, y0) is a point on the line. Substituting the point given and the slope 1/3, we get y + 3 = (x - -9)/3 = (x+9)/3. Adding three to both sides: y = (x + 9)/3 + 9/3 = (x + 9 + 9)/3 = (x + 18)/3. So the equation we seek is y = (x + 18)/3 or y= x/3 + 6

2.The slope is -2/3, so the slope of our new line will be 3/2. It has to pass through (-8,3) so y-3 = 3(x+8)/2 or y = 3(x+10)/2

3. The slope is -1, so the slope of our new line will be 1. It has to pass through (3,-6) so y+6=(x-3) or y = x - 9

Comments

1. You can just switch the numbers in front of x and y and make one of them negative. i.e. instead of the original equation 3x+1y=5 write 1x-3y=...  Then for the Right hand side of the equation, write the same thing as the Left hand side but replace the x with -9 and y with 3 to get

1x-3y=1(-9)-3(3)=-9-9=-18. That is x-3y=-18 is the answer.

2. For question 2, you have to rewrite it in the form of question 1 in order to use this method. this will be 2x+3y=-8. so write

3x-2y=3(-3)-2(3)=-9-6=-15 so 3x-2y=-15 is your answer.

3. This is similar to the form of question 1. Instead of 1x+1y=-4 write

1x-1y=1(3)-1(-6) =3+6=9 so you have x-y=9 as answer.

                            

 

Back in College i found a simpler ways of solving equations. If the you need any further help feel free to ask.