Sure! Let's work through the steps to find the equation of the line in slope-intercept form, [tex]\( y = mx + b \)[/tex], that passes through the points [tex]\((-1, 2)\)[/tex] and [tex]\((6, 3)\)[/tex].
### Step-by-Step Solution
#### Step 1: Choose [tex]\((x_1, y_1)\)[/tex]
The first point is [tex]\((-1, 2)\)[/tex].
- [tex]\( x_1 = -1 \)[/tex]
- [tex]\( y_1 = 2 \)[/tex]
#### Step 2: Choose [tex]\((x_2, y_2)\)[/tex]
The second point is [tex]\((6, 3)\)[/tex].
- [tex]\( x_2 = 6 \)[/tex]
- [tex]\( y_2 = 3 \)[/tex]
#### Step 3: Calculate the slope ([tex]\(m\)[/tex])
The formula to calculate the slope [tex]\(m\)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substitute the values from the chosen points:
[tex]\[ m = \frac{3 - 2}{6 - (-1)} = \frac{1}{7} \approx 0.14285714285714285 \][/tex]
#### Step 4: Find the y-intercept ([tex]\(b\)[/tex])
To find the y-intercept [tex]\(b\)[/tex], we use the slope-intercept form of the equation [tex]\(y = mx + b\)[/tex]. We can rearrange this to solve for [tex]\(b\)[/tex]:
[tex]\[ b = y - mx \][/tex]
We can use either of the points to solve for [tex]\(b\)[/tex]. Let's use the point [tex]\((-1, 2)\)[/tex]:
[tex]\[ b = 2 - \left( \frac{1}{7} \times (-1) \right) \][/tex]
[tex]\[ b = 2 + \frac{1}{7} \][/tex]
[tex]\[ b \approx 2.142857142857143 \][/tex]
#### Step 5: Write the equation of the line
Now that we have both [tex]\(m\)[/tex] and [tex]\(b\)[/tex], we can write the equation of the line in slope-intercept form:
[tex]\[ y = mx + b \][/tex]
[tex]\[ y = \frac{1}{7} x + \frac{3}{7} \][/tex]
Or approximately:
[tex]\[ y = 0.14285714285714285 x + 2.142857142857143 \][/tex]
So, the equation of the line that passes through the points [tex]\((-1, 2)\)[/tex] and [tex]\((6, 3)\)[/tex] in slope-intercept form is:
[tex]\[ y = 0.14285714285714285 x + 2.142857142857143 \][/tex]
