Question

1. describe your process for finding slope. why does it not matter which points you choose to find it? now, consider the algebraic process for finding slope and use it to inform your responses.

Explanation:

Step1: Recall Slope Formula

The slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \( m=\frac{y_2 - y_1}{x_2 - x_1} \), where \( x_2
eq x_1 \).

Step2: Process to Find Slope

1. Identify two points \((x_1, y_1)\) and \((x_2, y_2)\) on the line.
2. Substitute the coordinates into the slope formula: calculate the difference in \( y \)-values (\( y_2 - y_1 \)) and the difference in \( x \)-values (\( x_2 - x_1 \)).
3. Divide the \( y \)-difference by the \( x \)-difference to get the slope.

Step3: Why Points Don't Matter

A line has a constant slope (for non - vertical lines). Let's take three points \((x_1,y_1)\), \((x_2,y_2)\), \((x_3,y_3)\) on a line. The slope between \((x_1,y_1)\) and \((x_2,y_2)\) is \( m_1=\frac{y_2 - y_1}{x_2 - x_1} \), and between \((x_2,y_2)\) and \((x_3,y_3)\) is \( m_2=\frac{y_3 - y_2}{x_3 - x_2} \). Since the line is straight, the ratio of the change in \( y \) to the change in \( x \) is constant. Mathematically, if we consider the slope between \((x_1,y_1)\) and \((x_3,y_3)\), \( m_3=\frac{y_3 - y_1}{x_3 - x_1}=\frac{(y_3 - y_2)+(y_2 - y_1)}{(x_3 - x_2)+(x_2 - x_1)} \). By the property of fractions (if \( \frac{a}{b}=\frac{c}{d}=k \), then \( \frac{a + c}{b + d}=k \) when \( b + d
eq0 \)), we can show that \( m_1 = m_2=m_3 \). So, no matter which two points we choose on the line, the slope will be the same.

Answer:

To find the slope: 1. Identify two points \((x_1,y_1)\) and \((x_2,y_2)\) on the line. 2. Use the formula \( m=\frac{y_2 - y_1}{x_2 - x_1} \) (with \( x_2
eq x_1 \)) to calculate the slope. It does not matter which points are chosen because a line has a constant slope (for non - vertical lines), and the ratio of the change in \( y \) to the change in \( x \) between any two points on the line is the same.