Question

Write the equation of the line that is perpendicular to the line shown in the graph and that will pass through the point (5,-2).

Answer 1

`\boxed{y = (2)/(5)x - 4}`

Explanation:

Let's find the equation for the line

Take two distinguishable points on the graph.

(0, 4) and (2, -1) are two points with integer values for coordinates through which the line passes

The equation of a line in slope-intercept form is
y = mx + b

where
m = slope
b = y-intercept

Slope m = rise/run

rise = difference in y values of any two chosen points

run = difference in corresponding x values for the same chosen points

Taking the two points (0, 4) and (2, -1)

rise = -1 - 4 = -5

run = 2 - 0 =2

So slope = - 5/2

The y-intercept of this line is 4, - this is the y-value where the line crosses the y-axis and corresponds to the value of y when x = 0

The equation of the line is
y = mx + b

`\rightarrow y = -(5)/(2)x + 4 \;\;\cdots \cdots (1)\\\\`

A line perpendicular to this line will have a slope that is the negative of the reciprocal of the slope.

The product of the two slopes will be -1

Reciprocal of -5/2 = -2/5
Negative of this is -(-2/5) = 2/5

So slope of perpendicular line = 2/5

Equation of line is

`y = (2)/(5)x + b`

To find b plug in x, y values for (5, -2) and solve for b

`-2 = (2)/(5)\cdot 5 + b\\\\`

`-2 = 2 + b\\\\-4 = b\\\\b = -4\\`

`\textrm{Equation of the line is}\\\\\boxed{y = (2)/(5)x - 4}`