An illustration of a coordinate plane with four quadrants shows the x-axis ranging from negative 9 to 9 and the y-axis ranging from negative 11 to 11 in increments of 1. A solid upward line with arrows at both ends is plotted on the graph. The line passes through the plotted points left parenthesis 0 comma negative 2 right parenthesis and left parenthesis 1 comma 3 right parenthesis.

Question

Use the graph to derive the equation of the line in slope-intercept form.

Answer 1

To derive the equation of the line in slope-intercept form (which is \(y = mx + b\)), we need to determine the slope \(m\) and the y-intercept \(b\).

1. Finding the slope \(m\): The slope \(m\) is calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Using the points \((0, -2)\) and \((1, 3)\):

   • \( (x_1, y_1) = (0, -2) \)
   • \( (x_2, y_2) = (1, 3) \)

   Plugging in the values, we have: \[ m = \frac{3 - (-2)}{1 - 0} = \frac{3 + 2}{1} = \frac{5}{1} = 5 \]

2. Finding the y-intercept \(b\): The y-intercept \(b\) is the value of \(y\) when \(x = 0\). From the point \((0, -2)\), we can see that: \[ b = -2 \]

3. Writing the equation: Now that we have the slope and y-intercept, we can substitute these values into the slope-intercept form: \[ y = mx + b \] Substituting \(m = 5\) and \(b = -2\): \[ y = 5x - 2 \]

Therefore, the equation of the line in slope-intercept form is: \[ y = 5x - 2 \]