Problem 2 - The Calculus with Analytic Geometry

Question

In Exercises 1 through 4 , find the slope of the line through the given points. $$ (5,2),(-2,-3) $$

Step-by-Step Solution

Answer

The slope is \( \frac{5}{7} \).

Step 1 - Recall the Slope Formula

The slope of a line passing through two points can be found using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Here, \bm\bm is the slope, and (x_1, y_1) and (x_2, y_2) are the coordinates of the two points.

Step 2 - Identify the Coordinates

Identify the coordinates from the given points. Here, (x_1, y_1) = (5, 2) and (x_2, y_2) = (-2, -3)

Step 3 - Substitute the Values into the Formula

Substitute the coordinates into the slope formula: \[ m = \frac{-3 - 2}{-2 - 5} \]

Step 4 - Simplify the Expression

Perform the arithmetic operations for the numerator and the denominator: \[ m = \frac{-5}{-7} \]

Step 5 - Simplify the Fraction

Simplify the fraction if possible: \[ m = \frac{5}{7} \]

Key Concepts

slope formula

The slope formula is fundamental when it comes to understanding how steep a line is on a graph. The slope, denoted as \( m \), is calculated by finding the 'rise' over the 'run' between two distinct points on a line. Basically, how much the line goes up or down (rise) for every unit it goes left or right (run). The general formula is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] To compute this, you need two points: (\( x_1, y_1 \)) and (\( x_2, y_2 \)).
Understanding this formula makes analyzing and graphing linear equations much simpler! So remember: rise over run.

coordinates

Coordinates are pairs of numbers that describe specific points on a graph. Each coordinate pair ( x, y ) tells you the exact position of a point. For example, in the problem, the given points are (5, 2) and (-2, -3).
The first number in each pair is the x-coordinate (horizontal position), and the second is the y-coordinate (vertical position).
By identifying coordinates correctly, you can easily place points on a graph and use them to calculate the slope. Coordinates guide you visually and numerically to solve linear equations.

simplifying fractions

Simplifying fractions is an essential step in finding the slope. Once you substitute the coordinates into the slope formula, you'll likely get a fraction that needs simplification. Take the result from the problem: \[ m = \frac{-5}{-7} \] Notice that both the numerator and the denominator are negative. Dividing two negative numbers results in a positive number. So, \[ m = \frac{5}{7} \] This is your simplified slope.
Simplifying fractions helps in making the final answer more comprehensible and useful for further calculations.

arithmetic operations

Arithmetic operations are basic math functions like addition, subtraction, multiplication, and division. To find the slope, it's crucial to perform these operations accurately.
For example, from the given exercise:

• First, subtract the y-coordinates: -3 - 2 = -5.
• Second, subtract the x-coordinates: -2 - 5 = -7.
• Finally, divide these results: -5 divided by -7 equals 5/7.

Being comfortable with these operations ensures that you can follow through with any linear equations or similar math problems with ease.