Problem 1 - Elementary Algebra

Question

For exercises 1-8, find the slope of the line that passes through the given points. $$ (9,15)(18,42) $$

Step-by-Step Solution

Answer

The slope of the line is 3.

Step 1: Identify the given points

The given points are (9, 15) and (18, 42). Label these points as \(x_1, y_1\) and \(x_2, y_2\). Here, \(x_1 = 9\), \(y_1 = 15\), \(x_2 = 18\), and \(y_2 = 42\).

Step 2: Use the slope formula

The formula for the slope of a line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \].

Step 3: Substitute the values into the formula

Substitute \(x_1 = 9\), \(y_1 = 15\), \(x_2 = 18\), and \(y_2 = 42\) into the formula to get: \[ m = \frac{42 - 15}{18 - 9} \]

Step 4: Perform the calculations

Calculate the difference in the y-coordinates and the difference in the x-coordinates: \[ y_2 - y_1 = 42 - 15 = 27 \] \[ x_2 - x_1 = 18 - 9 = 9 \] Substitute these values back into the slope formula: \[ m = \frac{27}{9} \]

Step 5: Simplify the fraction

Simplify \[ \frac{27}{9} \] to get \[ m = 3 \].

Key Concepts

Coordinate Geometry

Coordinate geometry is a branch of mathematics that allows us to represent geometric shapes and analyze their properties using algebraic equations.
Point locations are described using coordinates, which are pairs of numbers in the format (x, y).
The first number is the x-coordinate, representing the horizontal position, and the second number is the y-coordinate, representing the vertical position.
By using these coordinates, we can easily describe points, lines, and shapes on a coordinate plane.
In this exercise, we were given two points (9, 15) and (18, 42). These coordinates help us determine the slope of the line passing through them.

Slope Formula

The slope of a line measures its steepness and direction.
The slope formula helps us find this measure using the coordinates of two points ( \(x_1, y_1\) ) and ( \(x_2, y_2\) ).
The formula for the slope \( m \) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Here,

• \( y_2\): y-coordinate of the second point
• \( y_1\): y-coordinate of the first point
• \( x_2\): x-coordinate of the second point
• \( x_1\): x-coordinate of the first point

.
By substituting the coordinates of the points into the formula, we can find the slope.

Simplifying Fractions

Simplifying fractions is an important step in many mathematical problems.
It involves reducing fractions to their simplest form by dividing both the numerator (top number) and the denominator (bottom number) by their greatest common divisor (GCD).
In our exercise, we calculated the slope to be \( \frac{27}{9} \).
To simplify, divide both 27 and 9 by their GCD, which is 9:
\[ \frac{27 ÷ 9}{9 ÷ 9} = \frac{3}{1} = 3 \].
This makes our final slope value much clearer and concise.

Algebra

Algebra involves using symbols and letters to represent numbers and expressions, making it easier to generalize and solve mathematical problems.
In the context of finding the slope of a line, algebra helps us manipulate equations and substitute values.
The slope formula is a perfect example of using algebra to solve a problem:

• Identify the given points and label them as \(x_1, y_1\) and \(x_2, y_2\).
• Substitute these values into the slope formula \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
• Simplify the resulting fraction to get the final slope.

These steps show how algebra efficiently translates geometric problems into numeric solutions.