91Ó°ÊÓ

In coordinate geometry, the slope of a line is a measure of its steepness. Knowing how to find the slope is essential when dealing with line equations. The formula for the slope, denoted as \(m\), can be remembered by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Here, \((x_1, y_1)\) and \((x_2, y_2)\) are two points on the line. This formula calculates the 'rise' over 'run', showing how much the line goes up or down for each unit it goes across. For example, if we have two points like \((3.7, -1.05)\) and \((-2.16, 4.9)\), substituting them into the slope formula, we get: \[ m = \frac{4.9 - (-1.05)}{-2.16 - 3.7} \] This will help find the slope by determining the difference between the y-coordinates divided by the difference between the x-coordinates.

Coordinate geometry allows us to use algebraic techniques to solve geometry problems. It involves the use of the coordinate plane, a grid defined by a horizontal (x-axis) and a vertical (y-axis) line. Each point on this plane has an ordered pair of coordinates \((x, y)\) that denote its position. When working with points like \((3.7, -1.05)\) and \((-2.16, 4.9)\), we can visualize them on this grid. This helps in comprehending how the slope formula works in a two-dimensional space. By moving from point \( A (x_1, y_1) \) to point \( B (x_2, y_2) \), we can see the change in the y-values and x-values, which contributes to our understanding of the line's steepness—its slope.

When dealing with lines, it's important to know their equations. They are often written in the slope-intercept form: \[ y = mx + b \] Here, \(m\) is the slope and \(b\) is the y-intercept (where the line crosses the y-axis). Once we know the slope using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \), we can plug it into this form if we know one of the points on the line. In our case, with the slope \( m \approx -1.02 \) from the points \((3.7, -1.05)\) and \((-2.16, 4.9)\), finding the equation is straightforward. If needed, we can substitute one of the points into the equation to find \(b\).

Accurate numerical computation is key when calculating the slope of a line, especially when working with decimals. Let's break it down: First, identify your points: \((x_1, y_1) = (3.7, -1.05)\) and \((x_2, y_2) = (-2.16, 4.9)\). Next, calculate the numerator: \[ y_2 - y_1 = 4.9 - (-1.05) = 4.9 + 1.05 = 5.95 \] Then, find the denominator: \[ x_2 - x_1 = -2.16 - 3.7 = -5.86 \] Finally, compute the slope: \[ m = \frac{5.95}{-5.86} \approx -1.02 \] Breaking down each step ensures clarity and prevents mistakes. Consider rounding your results to prevent miscalculations, such as rounding to the nearest hundredth to get \(-1.02\).