To write the equation of the line that passes through the points [tex]\((-1, 2)\)[/tex] and [tex]\((6, 3)\)[/tex] in slope-intercept form, we will go through the steps sequentially:
### Step 1: Choose [tex]\((x_1, y_1)\)[/tex]
We select the point [tex]\((-1, 2)\)[/tex], so:
[tex]\[ x_1 = -1, \quad y_1 = 2 \][/tex]
### Step 2: Identify [tex]\(x_2\)[/tex] and [tex]\(y_2\)[/tex]
[tex]\[ x_2 = 6, \quad y_2 = 3 \][/tex]
### Step 3: Calculate the slope [tex]\(m\)[/tex]
The slope of a line passing through 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:
[tex]\[ m = \frac{3 - 2}{6 - (-1)} \][/tex]
[tex]\[ m = \frac{1}{7} \][/tex]
[tex]\[ m = 0.14285714285714285 \][/tex]
### Step 4: Determine the y-intercept [tex]\(b\)[/tex]
Using the slope-intercept form of the equation of a line [tex]\(y = mx + b\)[/tex], we substitute one of the points [tex]\((-1, 2)\)[/tex] and the slope [tex]\(m = 0.14285714285714285\)[/tex] to solve for the y-intercept [tex]\(b\)[/tex]:
[tex]\[ 2 = 0.14285714285714285(-1) + b \][/tex]
[tex]\[ 2 = -0.14285714285714285 + b \][/tex]
[tex]\[ b = 2 + 0.14285714285714285 \][/tex]
[tex]\[ b = 2.142857142857143 \][/tex]
### Step 5: Write the equation of the line
Now, using the slope [tex]\(m\)[/tex] and y-intercept [tex]\(b\)[/tex], we write the equation in slope-intercept form:
[tex]\[ y = 0.14285714285714285x + 2.142857142857143 \][/tex]
Thus, the equation of the line that passes through the points [tex]\((-1, 2)\)[/tex] and [tex]\((6, 3)\)[/tex] is:
[tex]\[ y = 0.14285714285714285x + 2.142857142857143 \][/tex]
