Question

What is the slope of the line that passes through the coordinates \((1,-7)\) and \((8,5)\) ? (A) \(-\frac{2}{7}\) (B) \(-\frac{7}{2}\) (C) \(\frac{12}{7}\) (D) \(\frac{7}{12}\)

Short answer

The slope of the line that passes through the coordinates $(1, -7)$ and $(8, 5)$ is \(\boxed{\frac{12}{7}}\).

Step-by-step solution

Identifying the Coordinates

We are given the coordinates \((1, -7)\) and \((8, 5)\). Let's set \((x1, y1) = (1, -7)\) and \((x2, y2) = (8, 5)\).

Applying the Slope Formula

Using the slope formula, it's time to plug in the coordinates to find the slope of the line: \[ m = \frac{y2 - y1}{x2 - x1} = \frac{5 - (-7)}{8 - 1} \]

Simplifying the Expression

Now, let's simplify the expression: \[ m = \frac{5+7}{8-1} = \frac{12}{7} \]

Comparing to Answer Choices

The calculated slope, \(\frac{12}{7}\), matches answer choice (C). So, the slope of the line that passes through the coordinates \((1, -7)\) and \((8, 5)\) is \(\boxed{\frac{12}{7}}\).

Key concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a method of geometry using a coordinate system to describe geometrical shapes and their properties. It is a tool that helps us understand and solve problems involving shapes, lines, and figures in a coordinate plane.
Understanding coordinate geometry is key when exploring how different points connect to form lines or curves. When points on a plane are given in coordinates like \(x_1, y_1\) and \(x_2, y_2\), you can calculate various attributes, such as the distance between them or the slope of the line connecting them.

• Coordinates: These are the ordered pairs \( (x, y) \) which provide a precise position on the plane.
• Line: A line in coordinate geometry is a straight path connecting two points, extending infinitely in both directions.
• Plane: A flat, two-dimensional surface where every point is defined by a pair of coordinates.

Coordinate geometry extends beyond simple point locations. It includes determining slopes, as well as forming equations to represent the relationships between geometric entities.

Slope Formula

The slope of a line is an important concept in coordinate geometry. It represents the steepness and direction of a line. When you know two points on a line, you can use the slope formula to find out how much the line tilts up or down.
The mathematical formula for slope, usually denoted as \(m\), is given by:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]This formula calculates the vertical change (rise) between two points divided by the horizontal change (run) between the same points. Here’s how it works:

• Rise: The change in the y-values (i.e., \(y_2 - y_1\)).
• Run: The change in the x-values (i.e., \(x_2 - x_1\)).

In our original problem, using points \(1, -7\) and \(8, 5\), we substitute into the slope equation as follows: The rise is \(5 - (-7)\) which simplifies to 12, and the run is \(8 - 1\) which is 7.
Hence, the slope \(m\) is \(\frac{12}{7}\). Understanding and applying this formula helps not only in geometry problems but also in real-world situations involving slopes and rates of change.

Linear Equations

A linear equation is fundamental in both algebra and geometry. These equations describe a straight line and have the general form \(Ax + By + C = 0\), but when expressed in a more familiar slope-intercept form, it looks like \(y = mx + b\). This form emphasizes:

• m: The slope of the line, representing its inclination.
• b: The y-intercept, the point where the line crosses the y-axis.

With our calculated slope of \(\frac{12}{7}\), if we let \(b\) be determined through substitution of any of the given points, we can describe the entire line with just an equation. It's essential to understand that
- Linear equations form the backbone of algebraic graphing, often used in programming, economics, and science.
- They represent constant rates of change, making them very useful when depicting real-world relationships and natural phenomena.
Studying linear equations gives a robust tool for modeling and solving complex problems, reflecting both direct and inverse relationships between variables.