Problem 15 - Understanding Elementary Algebra with Geometry

Question

Find the slope of the line passing through the given points. Round to the nearest hundredth where necessary. \(\left(1, \frac{1}{2}\right)\) and \(\left(\frac{3}{4}, 2\right)\)

Step-by-Step Solution

Answer

The slope of the line is \(-6\).

Step 1: Identify the Coordinates

Identify the given coordinates as \((x_1, y_1) = \left(1, \frac{1}{2}\right)\) and \((x_2, y_2) = \left(\frac{3}{4}, 2\right)\).

Step 2: Recall the Slope Formula

The slope \(m\) of a line passing through points \((x_1, y_1)\) and \((x_2, y_2)\) is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Step 3: Substitute the Coordinates

Substitute \((x_1, y_1)\) and \((x_2, y_2)\) into the slope formula: \[ m = \frac{2 - \frac{1}{2}}{\frac{3}{4} - 1} \]

Step 4: Evaluate the Numerator

Calculate the difference in the y-values: \[ 2 - \frac{1}{2} = \frac{4}{2} - \frac{1}{2} = \frac{3}{2} \]

Step 5: Evaluate the Denominator

Calculate the difference in the x-values: \[ \frac{3}{4} - 1 = \frac{3}{4} - \frac{4}{4} = -\frac{1}{4} \]

Step 6: Calculate the Slope

Using the values found for the numerator and the denominator, find the slope: \[ m = \frac{\frac{3}{2}}{-\frac{1}{4}} = \frac{3}{2} \times -4 = -6 \]

Step 7: Round the Result

Since the slope is exactly an integer, rounding is not necessary. The slope is \(-6\).

Key Concepts

slope formula

The slope of a line tells us how steep the line is. The formula to find the slope given two points, \((x_1, y_1)\) and \((x_2, y_2)\), is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] This formula is handy because it works for any two points on a straight line. Slope, denoted by \(m\), is the ratio of the change in y (vertical change) to the change in x (horizontal change). If you know the coordinates of any two points on a line, you can find the slope using this formula. Start by subtracting the y-values of the two points to get the numerator and the x-values to get the denominator. Simplify these differences if needed.

For example, given points \((1, \frac{1}{2})\) and \((\frac{3}{4}, 2)\), we use the slope formula as shown above. Constructing and simplifying our fractions correctly ensures we get the correct slope.

coordinate geometry

Coordinate geometry, or analytic geometry, is a method to study geometry using the coordinate system. It allows us to describe geometric properties and relationships algebraically. In the coordinate plane, any point can be represented by an ordered pair \((x, y)\), where \(x\) is the horizontal position and \(y\) is the vertical position.

When finding the slope, note how each point corresponds to \((x_1, y_1)\) and \((x_2, y_2)\). Recognizing these pairs is the first step. For our exercise, points \((1, \frac{1}{2})\) and \((\frac{3}{4}, 2)\) are translated directly into our slope formula. This approach harnesses both graphical and algebraic methods to solve problems involving lines and other shapes.

fraction arithmetic

Mastery of fraction arithmetic is crucial when dealing with slope calculations. The numerator and denominator in the slope formula often involves fractions. Knowing how to add, subtract, multiply, and divide fractions is essential. For the given points, when calculating \[ 2 - \frac{1}{2} = \frac{4}{2} - \frac{1}{2} = \frac{3}{2} \], you simplify by ensuring both terms have a common denominator.

Next, the expression \[ \frac{3}{4} - 1 \] involves converting 1 to a fraction with common denominators, \[ \ \frac{4}{4} \ \], to get \(-\frac{1}{4} \). Hence, fraction arithmetic is a vital skill.

Multiplying fractions is straightforward: \(\frac{\frac{3}{2}}{-\frac{1}{4}} = -\frac{3}{2} \times \frac{-4}{1} = -3 \times (-4) = -6 \). This step finalizes our slope calculation.

elementary algebra

Elementary algebra simplifies working with variables and known numbers, helping us solve equations or find specific values like our slope. This concept includes basic operations (addition, subtraction, multiplication, division) with algebraic expressions.

In slope calculations, we substitute known values into the formula: \[ m = \frac{2 - \frac{1}{2}}{\frac{3}{4} - 1} \]. By systematically substituting and simplifying, each step becomes less overwhelming. Here, algebra aids in clarifying seemingly complex manipulations.

Additionally, interpreting and simplifying negative and positive results correctly is another key algebraic skill. For instance, recognizing \(-6\) as our resulting slope, where the negative sign indicates the line descends from left to right, consolidates basic algebraic comprehension.