91Ó°ÊÓ

Coordinates provide a way to identify a point's position in a space, often in a 2-dimensional plane. In our problem, we are working with two such pairs of coordinates: \((-13, 58)\) and \((12, -34)\). Each of these pairs consists of an \(x\)-coordinate and a \(y\)-coordinate. Here's how to interpret them:

• The first number in the pair, \(x\), indicates the horizontal position from the origin of the Euclidean plane.
• The second number, \(y\), indicates the vertical position.

Coordinates are fundamental in understanding geometry and algebra as they help define the exact location of points in space. This makes them crucial when calculating the slope or solving other mathematical problems involving the positioning of points.

Understanding the slope formula is essential for determining the steepness or incline of a line connecting two points. The general formula for the slope \(m\) between points \((x_1, y_1)\) and \((x_2, y_2)\) is:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]In simple terms, the slope is the ratio between the vertical change and horizontal change between two points:

• The numerator \(y_2 - y_1\) reflects how much the \(y\)-values change.
• The denominator \(x_2 - x_1\) reflects how much the \(x\)-values change.

Slopes can be positive, negative, zero, or undefined, each describing different line behaviors:

• Positive slope: Line rises as it moves right.
• Negative slope: Line falls as it moves right.
• Zero slope: Horizontal line.
• Undefined slope: Vertical line.

In our exercise, substituting the given coordinates into the formula allows us to calculate the slope, illustrating the process of determining how steep a line segment between these two points is.

Algebra plays a key role in solving problems related to slopes and coordinates. In the given exercise, algebraic manipulation is essential to find the correct slope. Let's delve deeper into the algebraic steps:First, recognize the coordinates of the two points and plug them into the slope formula:\[ m = \frac{-34 - 58}{12 - (-13)} \]Applying basic algebraic principles, perform subtraction in the numerator:\(-34 - 58 = -92\) results in the total change in \(y\).Next, in the denominator, handle the removal of the negative sign in \(x_2 - (-x_1)\):\(12 - (-13)\) transforms to \(12 + 13 = 25\).Finally, divide to find the slope:\[ m = \frac{-92}{25} \]This calculation shows the application of algebraic operations such as subtraction, addition, and fraction simplification, giving us the slope in its simplest form \(\frac{-92}{25}\). Algebra's rule-following nature ensures each operation leads us step-by-step to an accurate and meaningful result.