Problem 42 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the general form of the equation of the line that passes through the two points. \((0,-6),(-2,10)\)

Step-by-Step Solution

Answer

The general form of the equation of the line is 16x + y + 6 = 0.

Step 1: Calculate the Slope

You begin by finding the slope of the line, m. Use the formula m = \((y2-y1)/(x2-x1)\) where (x1 , y1) and (x2 , y2) are the given points. So the slope m = \((10 - (-6))/(-2 - 0) = -16\).

Step 2: Use the Point-Slope Form of the Line Equation

The point-slope form of a line's equation is y - y1 = m(x - x1). Insert the slope and one of the points. Here's the equation using the point (0,-6): y - (-6) = -16(x - 0).

Step 3: Convert to General Form

Now put the equation in general form, ax + by + c=0. Simplify y + 6 = -16x, this gives 16x + y + 6 = 0 which is the general form of the line.

Key Concepts

Slope Calculation

Understanding the slope calculation is crucial for anyone studying algebra and geometry, as it describes the steepness and direction of a line. To calculate the slope, often represented by the letter 'm', you can use the simple formula:
\[ m = \frac{(y_2-y_1)}{(x_2-x_1)} \]
This formula takes two points that the line passes through, typically denoted as \((x_1, y_1)\) and \((x_2, y_2)\). By subtracting the y-coordinates and dividing by the difference of the x-coordinates, you find the rate at which the line rises or falls as it moves from left to right. If the slope is positive, the line inclines upwards; if negative, it declines. A zero slope means the line is horizontal, and an undefined slope (division by zero) indicates a vertical line.

Point-Slope Form

Transitioning from slope to the actual equation of a line, the point-slope form comes handy, especially when we have a point through which the line passes and its slope. The point-slope formula is given by:
\[ y - y_1 = m(x - x_1) \]
'h4' In a Practical Scenario
To use this form, you need a single point \((x_1, y_1)\) and the slope 'm'. By substituting these values into the formula, you'll have an equation that represents the line. For example, given the point \((0, -6)\) and a slope of -16, the equation would be \(y + 6 = -16(x - 0)\). This format showcases the line’s characteristics clearly, with 'm' indicating the slope and \((x_1, y_1)\) signifying a point through which the line passes. It's an invaluable tool when graphing a line or writing its equation in other forms.

General Form Equation

Once we have the equation from the point-slope form, we might need to rewrite it into the general form, which is a standard way to express a line's equation. The general form is given as:
\[ ax + by + c = 0 \]
where 'a', 'b', and 'c' are real numbers. To convert from the point-slope to the general form, rearrange the terms such that x and y terms are on one side, and the constant is on the other side, and then bring all terms to one side so the constant is zero. Simplifying \(y + 6 = -16x\), we get the general form as \(16x + y + 6 = 0\). The general form is commonly used since it presents all variables on one side and zeroes out the other, simplifying certain types of analysis or solution methods.