Problem 20 - Understanding Elementary Algebra with Geometry

Question

Write an equation of the line satisfying the given conditions. Passing through \((-1,4)\) and \((2,-2)\)

Step-by-Step Solution

Answer

The equation of the line is \(y = -2x + 2\).

Step 1: Calculate the Slope

The slope of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Using the points \((x_1, y_1) = (-1, 4)\) and \((x_2, y_2) = (2, -2)\), substituting these into the slope formula gives \[ m = \frac{-2 - 4}{2 - (-1)} = \frac{-6}{3} = -2 \]

Step 2: Use Point-Slope Form

The equation of the line can be written using the point-slope form \[ y - y_1 = m(x - x_1) \] Taking one of the points \((x_1, y_1) = (-1, 4)\) and the slope \(m = -2\), plug in these values to get \[ y - 4 = -2(x + 1) \]

Step 3: Simplify to Slope-Intercept Form

Now, simplify the equation to get it into the slope-intercept form \(y = mx + b\). Expand and simplify the equation from Step 2: \[ y - 4 = -2x - 2 \] Adding 4 to both sides, we get \[ y = -2x + 2 \]

Step 4: Verify with the Second Point

To ensure the equation is correct, substitute the second point \((2, -2)\) into the equation \(y = -2x + 2\): \[ -2 = -2(2) + 2 = -4 + 2 = -2 \] Since both sides of the equation are equal, the equation is verified.

Key Concepts

Slope Calculation

Understanding how to calculate the slope of a line is crucial. The slope indicates how steep the line is, and whether it ascends or descends as you move from left to right. To find the slope between two points, use the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \] This formula represents the change in the y-coordinates (which is the rise) divided by the change in the x-coordinates (the run). In our example, we use the points \((-1, 4)\) and \((2, -2)\).
Plugging these values into the formula: \(
m = \frac{-2 - 4}{2 - (-1)} = \frac{-6}{3} = -2\)
Therefore, the slope \(m= -2\). This tells us the line falls 2 units for every 1 unit it moves to the right.

Point-Slope Form

The point-slope form of a line's equation is a valuable format for writing the equation of a line when you know one point on the line and the slope. The form is described as:
\[ y - y_1 = m(x - x_1) \] where \(m\) is the slope, and \( (x_1, y_1) \) is a specific point on the line. In the exercise, using the point \((-1, 4)\) and the slope \(-2\), we substitute these values into the formula:
\(
y - 4 = -2(x + 1)\)
This step allows us to fill in the known data to create the line equation in a simpler form before further simplifications.

Slope-Intercept Form

The slope-intercept form is probably the most commonly used and recognized form of a linear equation. It is represented as:
\[ y = mx + b \] In this format, \(m\) is the slope, and \(b\) is the y-intercept (the point where the line crosses the y-axis). After rearranging and simplifying the point-slope equation \(y - 4 = -2(x + 1)\), we get:
\(
y - 4 = -2x - 2\)
\( y = -2x + 2 \) Here, \(-2\) is the slope and \(2\) is the y-intercept. This form is easy to interpret and use for graphing lines, as you can immediately identify how steep the line is and where it crosses the y-axis.