Problem 72 - Algebra 1: Concepts and Skills

Question

Find the slope of the line that passes through the points. \((0,6)\) and \((8,0)\)

Step-by-Step Solution

Answer

The slope of the line that passes through the points \((0,6)\) and \((8,0)\) is \(-3/4\)

Step 1: Identify the Coordinates

Identify the assigned values for the coordinates of each point. For this problem, the first point, \((x_1, y_1)\), is \((0,6)\) and the second point, \((x_2, y_2)\), is \((8,0)\)

Step 2: Substitute in the Formula

Now substitute the identified values into the slope formula \((y_2 - y_1) / (x_2 - x_1)\). This gives \((0 - 6) / (8 - 0) = -6 / 8\)

Step 3: Simplify the Fraction

Simplify the fraction to find the slope of the line. Here, \(-6 / 8\) simplifies to \(-3 / 4\)

Key Concepts

Slope Formula

Understanding the slope formula is crucial for tackling problems in coordinate geometry. The slope of a line measures its steepness, often represented as 'rise over run'. This gives a clear picture of how much a line goes up or down as you move along it horizontally.

The slope formula is \( m = \frac{y_2 - y_1}{x_2 - x_1} \), where \( m \) stands for the slope, and \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of two distinct points on the line. The change in \( y \) (\( y_2 - y_1 \) ) is the 'rise', and the change in \( x \) (\( x_2 - x_1 \) ) is the 'run'.

For example, let's find the slope of the line through the points \( (0,6) \) and \( (8,0) \). Using our formula, we calculate the difference in \( y \) values (which is \( 0-6 = -6 \) ), and the difference in \( x \) values (which is \( 8-0 = 8 \) ). Substituting these into the slope formula gives us a slope of \( m = \frac{-6}{8} \) .

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a powerful branch of mathematics that uses algebraic equations to describe geometric concepts. It lays a bridge between algebra and geometry through the use of a coordinate system.

In this system, every point on a plane is represented by a pair of numbers, \( (x, y) \) , which are the coordinates of the point: \( x \) represents the horizontal distance from the origin, while \( y \) represents the vertical distance. Lines in this plane can be described by linear equations, and the slope of a line is one of the fundamental characteristics that can be determined through coordinate geometry.

When working with points like \( (0,6) \) and \( (8,0) \) to find the slope, you are applying principles of coordinate geometry. By plotting these points and using the slope formula, you effectively use algebra to understand and solve problems in a geometric space.

Simplifying Fractions

Simplifying fractions is a basic yet vital mathematical skill, which involves reducing a fraction to its simplest form. A fraction is simplified when the numerator and denominator have no common factors other than 1.

To simplify a fraction, you divide both the numerator and the denominator by their greatest common factor (GCF). You might sometimes hear this referred to as 'canceling down'. In the context of finding the slope, simplifying the fraction gives the most reduced form of slope, which is easier to interpret and use.

For example, the fraction \( \frac{-6}{8} \) from our slope calculation can be simplified by dividing both -6 and 8 by their GCF, which is 2. This results in the simplified fraction \( \frac{-3}{4} \). Simplifying fractions not only makes the numbers more manageable but also allows a better visualization of the slope's steepness in the case of geometric interpretations.