Problem 26 - Algebra 1: Concepts and Skills

Question

FINDING SLOPE Find the slope of the line that passes through the points. $$ (3,-4) \text { and }(9,4) $$

Step-by-Step Solution

Answer

The slope of the line that passes through the points (3,-4) and (9,4) is \(\frac{4}{3}\)

Step 1: Identify the given points

We are given the points (3,-4) and (9,4). We can identify these coordinates as \((x_1, y_1) = (3, -4)\) and \((x_2, y_2) = (9,4)\) respectively.

Step 2: Plug the points into the formula

Plugging the coordinates into the slope formula, we get \(m = \frac{4 -(-4)} {9 -3}\)

Step 3: Calculate the slope

Solving the expression in step 2 gives \(m = \frac{4 + 4}{9 - 3} = \frac{8}{6}\)

Step 4: Simplify the result

Simplifying the fraction \(\frac{8}{6}\) to its lowest terms, we get \(m = \frac{4}{3}\)

Key Concepts

Slope Formula

The slope of a line is a measure of its steepness and direction. In coordinate geometry, the slope is determined using the slope formula. The formula for calculating the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) on a line is: \[m = \frac{y_2 - y_1}{x_2 - x_1}.\]This formula is derived from the concept of "rise over run," where:

• "Rise" represents the change in \( y \)-coordinates, indicating vertical movement from one point to another.
• "Run" represents the change in \( x \)-coordinates, indicating horizontal movement.

For the given problem, finding the slope involves plugging the coordinates of the points into the formula. Make sure to perform the subtraction correctly to avoid errors in direction. It’s important to maintain the order of subtraction relative to each point so that results are consistent. This simple concept lies at the heart of understanding linear relationships in coordinate geometry.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses algebraic equations to describe geometric shapes and various properties of figures. Points on a plane are expressed in pairs known as coordinates, usually noted as \((x, y)\). These coordinates help locate points on the coordinate plane—a grid defined by horizontal (\(x\)-axis) and vertical (\(y\)-axis) lines.The concept of slope is central to coordinate geometry. It provides a quantitative way to represent how inclined a line is, based on precise point descriptions. By knowing two points on a line, the slope formula allows us to understand the general direction and steepness of the line:

• If the slope \( m \) is positive, the line inclines upwards as it moves from left to right.
• If \( m \) is negative, the line declines downwards.
• A slope of zero indicates a perfectly horizontal line.
• An undefined slope (division by zero) represents a vertical line.

Thus, calculating slopes using coordinate geometry unlocks insights into the behavior of linear relationships on a plane.

Simplifying Fractions

Simplifying fractions is an essential skill in both algebra and everyday math. It involves reducing a fraction to its smallest possible numerator and denominator while maintaining the same value. In the given context, the slope calculation yielded the fraction \(\frac{8}{6}\). Simplifying this fraction involves finding the greatest common divisor (GCD) of the numerator (8) and the denominator (6).

• Identify common factors of 8 and 6. The common factor here is 2.
• Divide both the numerator and the denominator by their GCD. \[ \frac{8 \div 2}{6 \div 2} = \frac{4}{3} \]

Thus, the fraction \(\frac{8}{6}\) simplifies to \(\frac{4}{3}\). Simplification is a critical step; it makes final answers clearer and easier to interpret, especially in more complex problems. Therefore, always check whether your fractions can be simplified for a cleaner, more comprehensible result.