91Ó°ÊÓ

The slope formula is a crucial concept in algebra and coordinate geometry. The slope, often denoted as \( m \), represents the steepness or incline of a line. Mathematically, the slope between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is calculated using the formula:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

This formula helps us understand how much \( y \) changes for a unit change in \( x \).

Useful tips:

• Identify the coordinates as \( (x_1, y_1) \) and \( (x_2, y_2) \) before applying the formula.
• Always subtract coordinates in the same order: \( y_2 - y_1 \) over \( x_2 - x_1 \).
• The slope can be positive, negative, zero, or undefined.

In our exercise, we substituted the coordinates into the slope formula and simplified it, allowing us to solve for the unknown variable \( a \).

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses algebra to study geometric properties.

The key idea is to represent geometric shapes with equations. In this exercise, points on the coordinate plane are defined by pairs of numbers (coordinates).

The focus here is on the line passing through two points, \((a-1, 2a)\) and \( (a, 6)\). Using the slope of this line, we can derive important information about the coordinates.

• Each point is in the form \(( x, y) \) where x and y are the horizontal and vertical coordinates, respectively.
• To determine the slope between any two points, apply the slope formula and use the given points.
• Understand the relationship between coordinates and algebraic properties through substitution.

This approach simplifies complex geometric problems into solvable algebraic equations. For our task, we breaking down the line's geometric properties using algebra revealed the necessary steps to determine the value of \(a\).

Algebraic solutions involve solving equations to find unknown variables. Here's how algebra was used in our exercise:

We started with the slope formula: \[ 5 = \frac{6 - 2a}{a - (a-1)} \] and simplified the denominator \(a - (a - 1) = 1 \):

The equation became:

\[ 5 = 6 - 2a \]

By isolating \(a\), we performed algebraic steps:

• Subtract 6 from both sides: \( 5 - 6 = -2a \) resulting in -1 = -2a.
• Divide by -2: \( a = \frac{1}{2} \).

When dealing with algebraic solutions:

• Always isolate the variable you need to solve for.
• Perform inverse operations to both sides of the equation.
• Check your solution by substituting back into the original equation.

This systematic approach is essential for solving various algebra problems and ensures accuracy throughout the process. In our case, isolating and solving for \(a\) using these algebraic steps led us to the correct answer.