Question

6.6 graphing in point - slope form
identify the slope and a point on the line for each equation.

1. $y + 2=-2(x - 2)$
2. $y + 1=-\frac{1}{3}(x + 1)$
3. $y - 3=-\frac{7}{4}(x - 1)$
4. $y - 4=-3(x + 3)$

graph the line for each equation.

1. $y - 4=\frac{1}{5}(x - 3)$
2. $y + 1=-\frac{2}{5}(x - 4)$
3. $y - 3=-2(x + 4)$
4. $y + 5 = 4(x + 2)$

Explanation:

Let's solve the problem of identifying the slope and a point on the line for each equation (using the point - slope form \(y - y_1=m(x - x_1)\), where \(m\) is the slope and \((x_1,y_1)\) is a point on the line).

Equation 1: \(y + 2=-2(x - 2)\)

Step 1: Recall the point - slope form

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.
We can rewrite the given equation \(y + 2=-2(x - 2)\) as \(y-(-2)=-2(x - 2)\).

Step 2: Identify the slope and the point

By comparing with \(y - y_1=m(x - x_1)\), we can see that the slope \(m=-2\) and the point \((x_1,y_1)=(2,-2)\).

Equation 2: \(y + 1=-\frac{1}{3}(x + 1)\)

Step 1: Recall the point - slope form

The point - slope form is \(y - y_1=m(x - x_1)\). We rewrite the given equation as \(y-(-1)=-\frac{1}{3}(x-(-1))\).

Step 2: Identify the slope and the point

By comparing with the point - slope form, the slope \(m =-\frac{1}{3}\) and the point \((x_1,y_1)=(-1,-1)\).

Equation 3: \(y - 3=-\frac{7}{4}(x - 1)\)

Step 1: Recall the point - slope form

The point - slope form is \(y - y_1=m(x - x_1)\).

Step 2: Identify the slope and the point

By comparing the given equation \(y - 3=-\frac{7}{4}(x - 1)\) with \(y - y_1=m(x - x_1)\), we find that the slope \(m=-\frac{7}{4}\) and the point \((x_1,y_1)=(1,3)\).

Equation 4: \(y - 4=-3(x + 3)\)

Step 1: Recall the point - slope form

Rewrite the equation \(y - 4=-3(x + 3)\) as \(y - 4=-3(x-(-3))\). The point - slope form is \(y - y_1=m(x - x_1)\).

Step 2: Identify the slope and the point

By comparing with the point - slope form, the slope \(m = - 3\) and the point \((x_1,y_1)=(-3,4)\).

Equation 5: \(y - 4=\frac{1}{5}(x - 3)\)

Step 1: Recall the point - slope form

The point - slope form is \(y - y_1=m(x - x_1)\).

Step 2: Identify the slope and the point

By comparing the given equation \(y - 4=\frac{1}{5}(x - 3)\) with \(y - y_1=m(x - x_1)\), we get that the slope \(m=\frac{1}{5}\) and the point \((x_1,y_1)=(3,4)\).

Equation 6: \(y + 1=-\frac{2}{5}(x - 4)\)

Step 1: Recall the point - slope form

Rewrite the equation as \(y-(-1)=-\frac{2}{5}(x - 4)\). The point - slope form is \(y - y_1=m(x - x_1)\).

Step 2: Identify the slope and the point

By comparing with the point - slope form, the slope \(m=-\frac{2}{5}\) and the point \((x_1,y_1)=(4,-1)\).

Equation 7: \(y - 3=-2(x + 4)\)

Step 1: Recall the point - slope form

Rewrite the equation as \(y - 3=-2(x-(-4))\). The point - slope form is \(y - y_1=m(x - x_1)\).

Step 2: Identify the slope and the point

By comparing with the point - slope form, the slope \(m=-2\) and the point \((x_1,y_1)=(-4,3)\).

Equation 8: \(y + 5=4(x + 2)\)

Step 1: Recall the point - slope form

Rewrite the equation as \(y-(-5)=4(x-(-2))\). The point - slope form is \(y - y_1=m(x - x_1)\).

Step 2: Identify the slope and the point

By comparing with the point - slope form, the slope \(m = 4\) and the point \((x_1,y_1)=(-2,-5)\).

Summary of Answers:

1. Slope: \(-2\), Point: \((2,-2)\)
2. Slope: \(-\frac{1}{3}\), Point: \((-1,-1)\)
3. Slope: \(-\frac{7}{4}\), Point: \((1,3)\)
4. Slope: \(-3\), Point: \((-3,4)\)
5. Slope: \(\frac{1}{5}\), Point: \((3,4)\)
6. Slope: \(-\frac{2}{5}\), Point: \((4,-1)\)
7. Slope: \(-2\), Point: \((-4,3)\)
8. Slope: \(4\), Point: \((-2,-5)\)

Answer:

Let's solve the problem of identifying the slope and a point on the line for each equation (using the point - slope form \(y - y_1=m(x - x_1)\), where \(m\) is the slope and \((x_1,y_1)\) is a point on the line).

Equation 1: \(y + 2=-2(x - 2)\)

Step 1: Recall the point - slope form

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.
We can rewrite the given equation \(y + 2=-2(x - 2)\) as \(y-(-2)=-2(x - 2)\).

Step 2: Identify the slope and the point

By comparing with \(y - y_1=m(x - x_1)\), we can see that the slope \(m=-2\) and the point \((x_1,y_1)=(2,-2)\).

Equation 2: \(y + 1=-\frac{1}{3}(x + 1)\)

Step 1: Recall the point - slope form

The point - slope form is \(y - y_1=m(x - x_1)\). We rewrite the given equation as \(y-(-1)=-\frac{1}{3}(x-(-1))\).

Step 2: Identify the slope and the point

By comparing with the point - slope form, the slope \(m =-\frac{1}{3}\) and the point \((x_1,y_1)=(-1,-1)\).

Equation 3: \(y - 3=-\frac{7}{4}(x - 1)\)

Step 1: Recall the point - slope form

The point - slope form is \(y - y_1=m(x - x_1)\).

Step 2: Identify the slope and the point

By comparing the given equation \(y - 3=-\frac{7}{4}(x - 1)\) with \(y - y_1=m(x - x_1)\), we find that the slope \(m=-\frac{7}{4}\) and the point \((x_1,y_1)=(1,3)\).

Equation 4: \(y - 4=-3(x + 3)\)

Step 1: Recall the point - slope form

Rewrite the equation \(y - 4=-3(x + 3)\) as \(y - 4=-3(x-(-3))\). The point - slope form is \(y - y_1=m(x - x_1)\).

Step 2: Identify the slope and the point

By comparing with the point - slope form, the slope \(m = - 3\) and the point \((x_1,y_1)=(-3,4)\).

Equation 5: \(y - 4=\frac{1}{5}(x - 3)\)

Step 1: Recall the point - slope form

The point - slope form is \(y - y_1=m(x - x_1)\).

Step 2: Identify the slope and the point

By comparing the given equation \(y - 4=\frac{1}{5}(x - 3)\) with \(y - y_1=m(x - x_1)\), we get that the slope \(m=\frac{1}{5}\) and the point \((x_1,y_1)=(3,4)\).

Equation 6: \(y + 1=-\frac{2}{5}(x - 4)\)

Step 1: Recall the point - slope form

Rewrite the equation as \(y-(-1)=-\frac{2}{5}(x - 4)\). The point - slope form is \(y - y_1=m(x - x_1)\).

Step 2: Identify the slope and the point

By comparing with the point - slope form, the slope \(m=-\frac{2}{5}\) and the point \((x_1,y_1)=(4,-1)\).

Equation 7: \(y - 3=-2(x + 4)\)

Step 1: Recall the point - slope form

Rewrite the equation as \(y - 3=-2(x-(-4))\). The point - slope form is \(y - y_1=m(x - x_1)\).

Step 2: Identify the slope and the point

By comparing with the point - slope form, the slope \(m=-2\) and the point \((x_1,y_1)=(-4,3)\).

Equation 8: \(y + 5=4(x + 2)\)

Step 1: Recall the point - slope form

Rewrite the equation as \(y-(-5)=4(x-(-2))\). The point - slope form is \(y - y_1=m(x - x_1)\).

Step 2: Identify the slope and the point

By comparing with the point - slope form, the slope \(m = 4\) and the point \((x_1,y_1)=(-2,-5)\).

Summary of Answers:

1. Slope: \(-2\), Point: \((2,-2)\)
2. Slope: \(-\frac{1}{3}\), Point: \((-1,-1)\)
3. Slope: \(-\frac{7}{4}\), Point: \((1,3)\)
4. Slope: \(-3\), Point: \((-3,4)\)
5. Slope: \(\frac{1}{5}\), Point: \((3,4)\)
6. Slope: \(-\frac{2}{5}\), Point: \((4,-1)\)
7. Slope: \(-2\), Point: \((-4,3)\)
8. Slope: \(4\), Point: \((-2,-5)\)