Problem 69 - Algebra 1

Question

MULTIPLE CHOICE What is the slope of the line passing through the points \((-3,4)\) and \((5,-11) ?\) \(\begin{array}{llll}\text { (A) \)\frac{-7}{8}\( } & \text { (B) \)\frac{-7}{2}\( } & \text { (C) \)\frac{15}{-8}\( }\end{array}\) (D) \(\frac{15}{8}\)

Step-by-Step Solution

Answer

\((C) \frac{15}{-8}\)

Step 1: Identify the given points

First, identify the coordinates of the two given points in the problem. In this case, they are \((-3,4)\) and \((5,-11)\).

Step 2: Apply the formula for slope

The formula for the slope of a line is \(m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\). Apply this formula substituting the values of \((x_1,y_1)\) and \((x_2,y_2)\) from the given coordinates. This gives \(m=\frac{-11-4}{5-(-3)}\).

Step 3: Simplify the expression

Simplify the expression in the numerator and the denominator to get \(m=\frac{-15}{8}\).

Key Concepts

Coordinate Geometry

Understanding coordinate geometry is crucial when dealing with problems involving points and lines in a plane. In coordinate geometry, each point is described using coordinates, which are a pair of numbers that specify its location relative to two reference lines, namely the x-axis and y-axis.
The x-coordinate indicates how far to move horizontally from the origin, while the y-coordinate shows the vertical movement. When finding the slope of a line, we identify two points on the line using their coordinates, which allows us to calculate the slope efficiently.
Coordinate geometry enables the translation of geometric problems into numerical problems, making it easier to apply algebraic methods and obtain solutions. These concepts are foundational in understanding further topics such as lines, circles, and other shapes in geometry.

Linear Equations

Linear equations are essential in understanding and describing the relationship between two variables on a coordinate plane. A linear equation can be in the form of \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept, or the point where the line crosses the y-axis.
The slope \(m\) represents the steepness of the line. It is calculated as the ratio of the vertical change to the horizontal change between two points on the line, as shown by the formula \(m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\).
Finding the slope is a crucial skill because it tells us how one variable changes with respect to another. Linear equations clearly demonstrate how two quantities are related, making them useful in analyzing patterns, trends, and even in making predictions.

Algebraic Expressions

Algebraic expressions play a vital role in solving geometry and algebra problems like finding the slope. In the slope formula \(m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\), the numerator and denominator are each algebraic expressions.
These expressions are derived from the coordinates of two points and require simplification to find the slope. Simplification involves basic operations like subtraction and division, helping students to develop fluency in manipulating expressions.
Understanding algebraic expressions and learning to simplify them is crucial in coordinate geometry, as it aids in transitioning from a problem statement to numerical results. This skill extends beyond finding slopes and is applicable in various areas of mathematics, reinforcing overall mathematical competence.