Problem 49 - Precalculus with Limits

Question

In Exercises 47-52, use a determinant to find an equation of the line passing through the points. \((-4, 3)\), \((2, 1)\)

Step-by-Step Solution

Answer

The equation of the line passing through the points (-4,3) and (2,1) is \(y = -1/3x + 5/3\).

Step 1: Calculation of Slope

Using the coordinates of the two points \((-4,3)\) and \((2,1)\), we can calculate the slope \(m\) of the line that passes through these points. The slope is calculated by the formula: \(m = (y2-y1) / (x2-x1)\), where \(x1,y1\) are the coordinates of the first point and \(x2,y2\) are the coordinates of the second point. Substituting the given points into this formula, we get \(m = (1-3) / (2-(-4)) = -2 / 6 = -1/3\).

Step 2: Calculation of y-intercept

Now that we have the slope, we can find the y-intercept \(c\) using the formula: \(c = y - mx\). We can use the coordinates of either of the two points. In this example, we'll use the point \((2,1)\). Using this point and the calculated slope, we substitute into the equation to find the y-intercept: \(c = 1 - (-1/3 * 2) = 1 - (-2/3) = 5/3\).

Step 3: Formulate the Equation of the Line

Now that we have calculated the slope \(m\) and the y-intercept \(c\), we can write down the equation of the line in the form \(y=mx+c\). Substituting \(m = -1/3\) and \(c = 5/3\), the equation of the line becomes: \(y = -1/3x + 5/3\).

Key Concepts

Slope Calculation

Understanding slope is crucial in coordinate geometry, as it measures how steep a line is. The slope is often denoted by the letter \(m\) and is defined by the change in \(y\) value divided by the change in \(x\) value between two points. When you have two points, \((x_1, y_1)\) and \((x_2, y_2)\), you can calculate the slope using the formula:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

For instance, if we have points \((-4, 3)\) and \((2,1)\), plug these into the formula to find: \(m = \frac{1-3}{2-(-4)} = \frac{-2}{6} = -\frac{1}{3}\).
This means every time \(x\) increases by 3, \(y\) decreases by 1. Thus, having a negative slope signifies that the line is descending.

Line Equation

Once you know the slope, forming a line equation is straightforward. The general equation for a line in slope-intercept form is:

• \( y = mx + c \)

Here, \(m\) represents the slope, and \(c\) is the y-intercept, which we will learn about shortly. Using our calculated slope \(m = -\frac{1}{3}\), we can continue to derive the equation of the line.
For example, substitute the slope and a point, let’s take point \((2, 1)\), into the equation using the formula for slope. This can be rearranged to help find the y-intercept \(y - mx = c\). This reformulated equation will guide you to find \(c\) and complete the line’s equation.

Y-Intercept

The y-intercept is the point at which the line crosses the y-axis. When \(x = 0\), the value of \(y\) represents the y-intercept, denoted by \(c\) in the equation \(y = mx + c\). It is the exact point where the line will intersect the y-axis and gives important information about where the line starts on the graph.To calculate the y-intercept, with a known slope \(m\), we take a point, say \((2, 1)\), and insert it into the rearranged line equation. Using \( m = -\frac{1}{3} \), substitute: \(1 - (-\frac{1}{3} \times 2) = 5/3\).
This result tells us the line crosses the y-axis at \(y = 5/3\). The y-value for this intercept gives a starting position for graphing and plotting the equation geometrically.

Coordinate Geometry

Coordinate geometry, or analytic geometry, allows us to explore geometric concepts through algebraic representation. By plotting points defined by their coordinates \((x, y)\), we can visualize lines, curves, and shapes on a plane.It’s a blend of algebra and geometry, linking equations to geometric figures. Consider it a bridge between abstract algebraic relationships and visual spatial awareness, making algebra visible.
The determination of a line’s equation via slope calculation and y-intercept is a prime example of using coordinate geometry efficiently. It simplifies complex geometric concepts, enabling you to analyze and interpret geometric shapes and patterns with precision.