Problem 20 - Algebra for College Students

Question

Find the slope of the line that contains each of the following pairs of points. \((3,4),(6,10)\)

Step-by-Step Solution

Answer

The slope of the line is 2.

Step 1 - Recall the Slope Formula

The slope formula for a line that passes through two points \((x_1, y_1)\text{ and }(x_2, y_2)\) is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1}\]

Step 2 - Substitute the Points into the Formula

Given points are \((3, 4)\text{ and }(6, 10)\). Substitute these values into the slope formula: \[ m = \frac{10 - 4}{6 - 3} \]

Step 3 - Simplify the Expression

Simplify the numerator and the denominator: \[ m = \frac{10 - 4}{6 - 3} = \frac{6}{3} \]

Step 4 - Calculate the Slope

Finish by simplifying the fraction: \[ m = 2 \]

Key Concepts

slope formula

When learning about the slope of a line, the first key concept is understanding the slope formula. The slope formula is a handy tool in coordinate geometry that helps us find how steep a line is. We can use this formula to determine the slope of a line connecting two points. The formula is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Here,

• m stands for the slope, which tells us how slanted our line is
• (x_1, y_1)\text{ and }(x_2, y_2) are the coordinates of the two given points

Understanding the formula is the first step, but applying it is also crucial. Plugging the points into the slope formula can show whether the slope is positive (the line goes upwards) or negative (the line goes downwards). This entire process is fundamental in coordinate geometry.

coordinate geometry

Coordinate geometry, or analytic geometry, is all about plotting points on a plane and examining the relationships between them. The plane is a flat surface, defined by the x-axis (horizontal) and y-axis (vertical). When we know the coordinates of two points, we can use the slope formula to find the slope of the connecting line.
For example, given the points \(\text{(3, 4)}\text{ and } \text{(6, 10})\), we can substitute these into the formula: \[ m = \frac{10 - 4}{6 - 3} \]
As we simplify, it illustrates the power of coordinate geometry to turn abstract points into a concrete slope that tells us about the relationship between those points.
Understanding coordinate geometry helps us understand complex relationships in algebra and calculus.

simplifying fractions

Simplifying fractions is a vital skill in many areas of math, including finding the slope. Once we apply the slope formula \[ m = \frac{y_2 - y_1}{x_2 - x_1} \], we often end up with a fraction. We need to simplify this fraction to its simplest form for a clear and correct slope.
As shown in the example:
Substituting \( \text{(3, 4) and (6, 10)} \) into the slope formula, we get:
\[ m = \frac{10 - 4}{6 - 3} = \frac{6}{3} \]
The fraction \( \frac{6}{3} \) simplifies to \( 2 \).
Simplifying a fraction means dividing the numerator and the denominator by their greatest common divisor (GCD). Knowing how to simplify fractions not only helps in calculating slopes, but also makes math problems more manageable.