Question

Write an equation in point-slope form of the line that passes through the given points. Then write the equation in slope intercept form.

1.) (5,1),(0,2)
2.) (-2,-3),(4,3)
3.) (-3,-2),(2,3)
4.) (2,5),(8,-7)

Answer 1

To find the equation of a line that passes through two given points, we need to first calculate the slope of the line using the formula:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Once we have the slope \( m \), we can use one of the points to write the equation in point-slope form:

\[
y - y_1 = m(x - x_1)
\]

Next, we can rearrange it into slope-intercept form \( y = mx + b \).

Let's go through each of the four cases.

### 1.) Points (5, 1) and (0, 2)

**Step 1: Calculate the slope \( m \)**

\[
m = \frac{2 - 1}{0 - 5} = \frac{1}{-5} = -\frac{1}{5}
\]

**Step 2: Write the equation in point-slope form using point (5, 1)**

\[
y - 1 = -\frac{1}{5}(x - 5)
\]

**Step 3: Write it in slope-intercept form**

\[
y - 1 = -\frac{1}{5}x + 1
\]
\[
y = -\frac{1}{5}x + 2
\]

### 2.) Points (-2, -3) and (4, 3)

**Step 1: Calculate the slope \( m \)**

\[
m = \frac{3 - (-3)}{4 - (-2)} = \frac{3 + 3}{4 + 2} = \frac{6}{6} = 1
\]

**Step 2: Write the equation in point-slope form using point (-2, -3)**

\[
y + 3 = 1(x + 2)
\]

**Step 3: Write it in slope-intercept form**

\[
y + 3 = x + 2
\]
\[
y = x - 1
\]

### 3.) Points (-3, -2) and (2, 3)

**Step 1: Calculate the slope \( m \)**

\[
m = \frac{3 - (-2)}{2 - (-3)} = \frac{3 + 2}{2 + 3} = \frac{5}{5} = 1
\]

**Step 2: Write the equation in point-slope form using point (-3, -2)**

\[
y + 2 = 1(x + 3)
\]

**Step 3: Write it in slope-intercept form**

\[
y + 2 = x + 3
\]
\[
y = x + 1
\]

### 4.) Points (2, 5) and (8, -7)

**Step 1: Calculate the slope \( m \)**

\[
m = \frac{-7 - 5}{8 - 2} = \frac{-12}{6} = -2
\]

**Step 2: Write the equation in point-slope form using point (2, 5)**

\[
y - 5 = -2(x - 2)
\]

**Step 3: Write it in slope-intercept form**

\[
y - 5 = -2x + 4
\]
\[
y = -2x + 9
\]

### Summary of Equations

1.) Point-Slope: \( y - 1 = -\frac{1}{5}(x - 5) \); Slope-Intercept: \( y = -\frac{1}{5}x + 2 \)

2.) Point-Slope: \( y + 3 = 1(x + 2) \); Slope-Intercept: \( y = x - 1 \)

3.) Point-Slope: \( y + 2 = 1(x + 3) \); Slope-Intercept: \( y = x + 1 \)

4.) Point-Slope: \( y - 5 = -2(x - 2) \); Slope-Intercept: \( y = -2x + 9 \)