Question

write an equation for the line that passes through each of the following points with the specified slope. write each equation in point - slope form, i.e. (y - y_1=m(x - x_1)).
(a) slope(=- 5)
point: ((2,-6))
(b) slope(=4)
point: ((-3,-11))
(c) slope(=\frac{3}{5})
point: ((9,4))

Explanation:

Part (a)

Step1: Recall point - slope formula

The point - slope form of a line is given by \(y - y_1=m(x - x_1)\), where \(m\) is the slope of the line and \((x_1,y_1)\) is a point on the line.
Here, \(m=- 5\) and the point \((x_1,y_1)=(2,-6)\).

Step2: Substitute values into formula

Substitute \(m = - 5\), \(x_1=2\) and \(y_1=-6\) into the point - slope formula \(y - y_1=m(x - x_1)\).
We get \(y-(-6)=-5(x - 2)\), which simplifies to \(y + 6=-5(x - 2)\).

Part (b)

Step1: Recall point - slope formula

The point - slope form of a line is \(y - y_1=m(x - x_1)\), where \(m\) is the slope and \((x_1,y_1)\) is a point on the line.
Here, \(m = 4\) and the point \((x_1,y_1)=(-3,-11)\).

Step2: Substitute values into formula

Substitute \(m = 4\), \(x_1=-3\) and \(y_1=-11\) into the formula \(y - y_1=m(x - x_1)\).
We have \(y-(-11)=4(x-(-3))\), which simplifies to \(y + 11=4(x + 3)\).

Part (c)

Step1: Recall point - slope formula

The point - slope form of a line is \(y - y_1=m(x - x_1)\), where \(m\) is the slope and \((x_1,y_1)\) is a point on the line.
Here, \(m=\frac{3}{5}\) and the point \((x_1,y_1)=(9,4)\).

Step2: Substitute values into formula

Substitute \(m=\frac{3}{5}\), \(x_1 = 9\) and \(y_1=4\) into the formula \(y - y_1=m(x - x_1)\).
We get \(y - 4=\frac{3}{5}(x - 9)\).

Answer:

s:
(a) \(y + 6=-5(x - 2)\)

(b) \(y + 11=4(x + 3)\)

(c) \(y - 4=\frac{3}{5}(x - 9)\)