Certainly! Let's find the equation of the line that passes through the points \((-7, -5)\), \((0, 1)\), and \((7, 7)\). The slope-intercept form of a line is given by \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
### Step 1: Calculate the Slope (\(m\))
The slope \(m\) of the line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Using the points \((-7, -5)\) and \((0, 1)\):
[tex]\[ x_1 = -7, \quad y_1 = -5 \][/tex]
[tex]\[ x_2 = 0, \quad y_2 = 1 \][/tex]
Substitute these values into the slope formula:
[tex]\[ m = \frac{1 - (-5)}{0 - (-7)} = \frac{1 + 5}{0 + 7} = \frac{6}{7} \][/tex]
So, the slope \(m\) is:
[tex]\[ m \approx 0.857 \][/tex]
### Step 2: Calculate the y-intercept (\(b\))
The y-intercept \(b\) can be found using the slope-intercept equation \(y = mx + b\). We can use any of the given points to find \(b\); let’s use the point \((0, 1)\).
Substitute \(x = 0\) and \(y = 1\) into the equation \(y = mx + b\):
[tex]\[ 1 = 0.857 \cdot 0 + b \][/tex]
Since \(0.857 \cdot 0 = 0\), we get:
[tex]\[ b = 1 \][/tex]
### Step 3: Write the Equation of the Line
Now we have found both the slope (\(m\)) and the y-intercept (\(b\)). Thus, the equation of the line in slope-intercept form is:
[tex]\[ y = 0.857x + 1 \][/tex]
So, the equation of the line that passes through the points \((-7, -5)\), \((0, 1)\), and \((7, 7)\) is:
[tex]\[ y = 0.857 \, x + 1 \][/tex]
This gives us a linear equation that accurately represents the relationship between [tex]\(x\)[/tex] and [tex]\(y\)[/tex] for the line passing through the given points.
