Bot GPT 4-o mini To find the equation of the line that passes through the points \((-2, 4)\) and \( (1, 13)\), we can use the slope-intercept form of a linear equation, which 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 a line through two points \((x_1, y_1)\) and \((x_2, y_2)\) can be calculated using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Using the points \((-2, 4)\) and \( (1, 13)\):
- \(x_1 = -2, y_1 = 4\)
- \(x_2 = 1, y_2 = 13\)
Substituting these values into the slope formula:
\[ m = \frac{13 - 4}{1 - (-2)} = \frac{9}{3} = 3 \]
Step 2: Use one of the points to find the y-intercept \(b\)
Now that we have the slope, we can use one of the points to find the y-intercept \(b\). We can use the point \((-2, 4)\):
\[ y = mx + b \] Substituting in \(m = 3\), \(x = -2\), and \(y = 4\):
\[ 4 = 3(-2) + b \] \[ 4 = -6 + b \] \[ b = 4 + 6 = 10 \]
Final equation
Now that we have \(m\) and \(b\), we can write the equation of the line:
\[ y = 3x + 10 \]
This is the equation that explains the line passing through the points \((-2, 4)\) and \( (1, 13)\).
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