Problem 1 - Algebra and Trigonometry

Question

Find the slope of the line through \(P\) and \(Q .\) \(P(0,0), Q(4,2)\)

Step-by-Step Solution

Answer

The slope of the line is \( \frac{1}{2} \).

Step 1: Identify the slope formula

The formula to find the slope (m) 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 2: Assign values to variables

From the coordinates, assign \(x_1 = 0\), \(y_1 = 0\), \(x_2 = 4\), and \(y_2 = 2\).

Step 3: Substitute values into the slope formula

Substitute the assigned values into the slope formula: \[ m = \frac{2 - 0}{4 - 0} \]

Step 4: Simplify the expression

Calculate the numerator: \(2 - 0 = 2\). Calculate the denominator: \(4 - 0 = 4\). So, \[ m = \frac{2}{4} \]

Step 5: Finalize the slope calculation

Simplify the fraction: \[ \frac{2}{4} = \frac{1}{2} \]. Therefore, the slope of the line is \( \frac{1}{2} \).

Key Concepts

Slope Formula

The slope formula is an essential tool in coordinate geometry used to determine the steepness or inclination of a line. It is pivotal for graphing linear relationships between two points on a plane. The formula for slope, represented by the letter \( m \), is:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

This expression indicates the rate of change along the vertical axis (\( y \)-axis) per unit change along the horizontal axis (\( x \)-axis). Think of it as the 'rise over run' method:

• Rise: The difference in the \( y \)-values \( (y_2 - y_1) \)
• Run: The difference in the \( x \)-values \( (x_2 - x_1) \)

The slope can be positive, negative, zero, or undefined. A positive slope means the line rises as it moves from left to right, and a negative slope indicates it descends. If the slope equals zero, the line is horizontal, whereas an undefined slope signals a vertical line. Understanding how to use the slope formula is crucial in analyzing and graphing linear equations.

Coordinate Geometry

Coordinate geometry allows us to visually understand algebraic equations by plotting them on a graph. It involves the study of geometric figures through the use of a coordinate plane, which is formed by two perpendicular number lines called axes.

• The horizontal line is the \( x \)-axis.
• The vertical line is the \( y \)-axis.

The intersection of these axes is called the origin, labeled as \((0, 0)\), which serves as a reference point for all other points on the plane. In coordinate geometry, each point is defined by an ordered pair \((x, y)\), which gives its exact location. When you want to find the slope of a line through two points, like \( P(0, 0) \) and \( Q(4, 2) \), you'll use these coordinates in the slope formula. This spatial representation makes it easier to grasp the concepts and the relationships between different mathematical elements. By applying coordinate geometry, we can solve for slopes, equations of lines, and even explore more advanced topics like curves and transformations.

Simplifying Fractions

Simplifying fractions is a critical skill in mathematics that helps make expressions more manageable. To simplify a fraction means to reduce it to its lowest possible terms. This process requires canceling out common factors in the numerator and denominator.Suppose you have a fraction like \( \frac{2}{4} \). The steps to simplify it are:

• Identify the greatest common divisor (GCD) of the numerator and denominator. In this case, the GCD of 2 and 4 is 2.
• Divide both the numerator and the denominator by their GCD. So, \( \frac{2}{4} \) becomes \( \frac{2 \div 2}{4 \div 2} \).
• The simplified fraction is \( \frac{1}{2} \).

A simplified fraction like \( \frac{1}{2} \) is easier to interpret. It often provides a clearer picture of ratios and proportions, which is especially useful in slope calculations where precision and simplicity matter. By mastering this process, you'll enhance your ability to solve a wide range of math problems effortlessly.