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Find the slope-intercept form of the equation of each line that passes through the point P and makes angle θ with the positive x-axis. P=(2,5),θ=135

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Answer:

To find the equation of a line passing through point P(2, 5) and making an angle of θ = 135 degrees with the positive x-axis, you can use the point-slope form of the equation of a line:

y - y₁ = m(x - x₁)

Where (x₁, y₁) is the point (2, 5), and m is the slope of the line.

First, let's find the slope (m) using the given angle θ. The slope of a line making an angle θ with the positive x-axis can be found using the tangent function:

m = tan(θ)

In this case, θ = 135 degrees. You can convert it to radians because most trigonometric functions expect angle values in radians:

θ (in radians) = (135°/180°) * π = (3/4) * π

Now, find the slope:

m = tan(3π/4)

The tangent of 3π/4 is -1. So, the slope (m) is -1.

Now, you can use the point-slope form with the values of (x₁, y₁) and m:

y - 5 = -1(x - 2)

Now, simplify:

y - 5 = -x + 2

To get the slope-intercept form (y = mx + b), isolate y:

y = -x + 2 + 5

y = -x + 7

So, the equation of the line passing through point P(2, 5) and making an angle of 135 degrees with the positive x-axis in slope-intercept form is:

y = -x + 7

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