- Calculate the slope of the line using the points \(J\) and \(K\): \(m = \frac{3 - (-5)}{-6 - (-4)} = -4\).
- Substitute the slope and one of the points into the slope-intercept form \(y = mx + b\) to solve for \(b\): \(-5 = -4(-4) + b\).
- Conclude that \(b = -21\).
- The final answer is \(\boxed{-21}\).
Explanation
- Calculate the slope of the line
To find the y-intercept \(b\) of the line passing through points \(J(-4, -5)\) and \(K(-6, 3)\), we start by finding the slope \(m\) of the line. The slope formula is given by:
\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
Substituting the coordinates of points \(J(-4, -5)\) and \(K(-6, 3)\) into the formula:
\[m = \frac{3 - (-5)}{-6 - (-4)} = \frac{3 + 5}{-6 + 4} = \frac{8}{-2} = -4\]
The slope of the line is \(-4\).
- Solve for the y-intercept b
Now that we have the slope \(m = -4\), we use the slope-intercept form \(y = mx + b\) to solve for the y-intercept \(b\). We can substitute the coordinates of point \(J(-4, -5)\) into the equation:
\[-5 = -4(-4) + b\]
\[-5 = 16 + b\]
Solving for \(b\):
\[b = -5 - 16 = -21\]
To ensure accuracy, we can verify this result using point \(K(-6, 3)\):
\[3 = -4(-6) + b\]
\[3 = 24 + b\]
\[b = 3 - 24 = -21\]
Both points yield the same y-intercept.
- Final Answer
The calculated value of the y-intercept \(b\) is \(-21\). This corresponds to the first option provided.
\[\boxed{-21}\]
Examples
The concept of finding the equation of a line is fundamental in various real-life scenarios, such as predicting future costs based on historical data. For instance, if you track the monthly cost of electricity (y) against your usage in kilowatt-hours (x), you can find a linear model \(y = mx + b\). The slope \(m\) represents the cost per unit of energy, and the y-intercept \(b\) represents the fixed monthly connection fee. Finding these parameters allows you to estimate your budget accurately for any given usage level.