Problem 44 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the general form of the equation of the line that passes through the two points. \((-4,12),(4,2)\)

Step-by-Step Solution

Answer

The general form of the equation of the line that passes through the points \((-4,12),(4,2)\) is \(x - y + 8 = 0\).

Step 1: Calculation of Slope

First, we find the slope of the line using the formula \(m = \frac{(y_2 - y_1)}{(x_2 - x_1)}\). Substituting the given points, we get \(m = \frac{(2 - 12)}{(4 - (-4))} = -1\). So, the slope of the line is -1.

Step 2: Find the equation of the line

Next, we use the point-slope form of the linear equation, which is \(y - y_1 = m(x - x_1)\). We can use either of the given points, let's use \((-4,12)\). Then the equation becomes \(y - 12 = -1(x + 4)\). Distribute -1 to get \(y - 12 = -x - 4\).

Step 3: General Form of the Line

Finally, we rearrange the equation to be in general form \(Ax + By + C = 0\). Adding \(x - y\) to both sides of the equation gets us \(x - y + 8 = 0\). Thus, the line that passes through the points \((-4,12)\) and \(4,2)\) can be represented by the general form equation \(x - y + 8 = 0\).

Key Concepts

Slope Calculation

Understanding the concept of slope is crucial when dealing with linear equations, as it describes how steep a line is. To calculate the slope of a line, the formula is \(m = \frac{(y_2 - y_1)}{(x_2 - x_1)}\). This formula essentially takes the change in the y-values, which is the vertical movement, divided by the change in x-values, the horizontal movement, between two points on the line. In the context of the exercise provided, the points \( (-4,12) \) and \( (4,2) \) were used to find the slope, resulting in \(m = -1\).

Slope can be positive, negative, zero, or undefined and this tells us the direction of the line. A positive slope means the line rises from left to right, negative indicates it falls, zero slope is a flat horizontal line, and an undefined slope is a vertical line. In this case, the negative slope indicates a downward trend from left to right.

Point-Slope Form

Once the slope is determined, the next step is to express the equation of the line. The point-slope form is particularly useful when we know a single point the line passes through and its slope. The form is given by \(y - y_1 = m(x - x_1)\), where \(m\) stands for slope, and \( (x_1, y_1) \) is the known point. For the given exercise, we used the point \( (-4,12) \) and the slope \(m = -1\) to derive the equation \(y - 12 = -1(x + 4)\).

This format is handy because it highlights the required information neatly: one point on the line and the slope. Expanding this equation out and simplifying, as in the solution, progresses us towards the more commonly seen linear equation forms.

General Form of Linear Equation

The general form of a linear equation is one of the standard ways to express the equation of a line, and it is usually written as \(Ax + By + C = 0\), where A, B, and C are real numbers. The final step in the exercise was converting the point-slope form into the general form, which in this case resulted in \(x - y + 8 = 0\). The process involved rearranging the terms and simplifying to get all components of the equation to one side, thus matching the general form structure.

The general form is widely used because it easily lends itself to analytic methods such as finding intercepts. In the given exercise, seeing the equation in general form makes it clear that both the x and y intercepts are numerically equal to 8, provided the equation is solved for those respective variables.