Question

The line y = mx + c is perpendicular to the line x+2y=10 and passes through the point (7,4). Find the values of m and c.

Answer 1

$1)y=\underbrace{m}_{k_{1}} x+c\\\\2)x+2y=10 \ \Rightarrow \ y=\underbrace{-0,5}_{k_{2}} x+5$

Условие перпендикулярности двух прямых :

$k_{1} \cdot k_{2}=-1$

Следовательно :

$m\cdot (-0,5)=-1\\\\m=\dfrac{-1}{-0,5}=2\\\\\boxed{m=2}$

По условию прямая y = mx + c проходит через точку

с координатами (7 ; 4) :

$y=mx+c\\\\m=2 \ \Rightarrow \ y=2x+c\\\\(7 \ ; \ 4) \ \Rightarrow \ x=7 \ , \ y=4\\\\4=2\cdot 7+c\\\\4=14+c\\\\c=4-14\\\\\boxed{c=-10}$

Answer 2

To find the values of m and c for the line y = mx + c that is perpendicular to the line x + 2y = 10 and passes through the point (7,4), you can follow these steps:

1. Determine the slope of the given line x + 2y = 10:

First, rewrite the equation in slope-intercept form (y = mx + b): x + 2y = 10 2y = -x + 10 y = (-1/2)x + 5

So, the slope of the given line is -1/2.

2. Determine the slope of the perpendicular line:

Since the line you're looking for is perpendicular to the given line, the slopes are negative reciprocals of each other. In other words, the slope of the perpendicular line is the negative reciprocal of -1/2, which is 2.

3. Use the point-slope form of a line to find the equation of the perpendicular line:

The point-slope form of a line is given by: y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line and m is the slope.

You are given the point (7,4) and the slope m = 2. Plug these values into the equation:

y - 4 = 2(x - 7)

4. Expand and simplify:

y - 4 = 2x - 14

5. Move -4 to the other side of the equation:

y = 2x - 14 + 4

6. Combine constants:

y = 2x - 10

Now, you have the equation of the line in the form y = mx + c, where m = 2 and c = -10. So, the values of m and c are:

m = 2 c = -10

The equation of the line that is perpendicular to x + 2y = 10 and passes through the point (7,4) is y = 2x - 10.