What is the equation of the line that pases through the points `(-3,1) and (1,3)` ?

Options

A. `y =-x+2`
B. `y =-x-2`
C. `y=x-2`
D. `y=x+2`

Explanation

To find the equation of the line that passes through the points \((-3, 1)\) and \((1, 3)\), we will follow these steps:

### Step 1: Identify the points
We have two points:
- Point 1: \((x_1, y_1) = (-3, 1)\)
- Point 2: \((x_2, y_2) = (1, 3)\)

### Step 2: Calculate the slope (m)
The formula for the slope \(m\) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

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

Substituting the values:

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

### Step 3: Use the slope-intercept form
The equation of a line in slope-intercept form is:

\[ y = mx + c \]

We already found \(m = \frac{1}{2}\). Now we can use one of the points to find \(c\). Let's use the point \((1, 3)\):

\[ 3 = \frac{1}{2}(1) + c \]

### Step 4: Solve for c
Now we solve for \(c\):

\[ 3 = \frac{1}{2} + c \]

Subtract \(\frac{1}{2}\) from both sides:

\[ c = 3 - \frac{1}{2} = \frac{6}{2} - \frac{1}{2} = \frac{5}{2} \]

### Step 5: Write the equation of the line
Now we have both \(m\) and \(c\):

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

### Step 6: Simplify the equation (optional)
If desired, we can multiply through by 2 to eliminate the fraction:

\[ 2y = x + 5 \]

Or, rearranging gives:

\[ x - 2y + 5 = 0 \]

### Final Answer
The equation of the line that passes through the points \((-3, 1)\) and \((1, 3)\) is:

\[ y = \frac{1}{2}x + \frac{5}{2} \]
or
\[ x - 2y + 5 = 0 \]

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