Problem 64 - Algebra 2

Question

Find the slope of the line through each pair of points. $$ \left(-\frac{1}{2},-\frac{1}{2}\right) \text { and }(-3,-4) $$

Step-by-Step Solution

Answer

\( m = \frac{7}{5} \)

Step 1: Identify the coordinates of the two points

Designate the first point as point A with coordinates \( A(-\frac{1}{2}, -\frac{1}{2}) \) and the second point as point B with coordinates \( B(-3, -4) \) .

Step 2: Use the slope formula

Apply the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \) by substituting the y-coordinates and x-coordinates of points A and B.

Step 3: Solve for the slope

After substitution, calculate the slope as \( m = \frac{-4 - (-\frac{1}{2})}{-3 - (-\frac{1}{2})} \) which simplifies to \( m = \frac{-4 + \frac{1}{2}}{-3 + \frac{1}{2}} \) or \( m = \frac{-\frac{7}{2}}{-\frac{5}{2}} \) which simplifies further to \( m = \frac{7}{5} \).

Key Concepts

Slope Formula

Understanding the slope formula is crucial when determining the steepness and direction of a line on the coordinate plane. The slope, denoted as 'm', can be found using the formula \[ m = \frac{{y_2 - y_1}}{{x_2 - x_1}} \]where \[ (x_1, y_1) \] and \[ (x_2, y_2) \]are coordinates of two distinct points on the line. The numerator \( y_2 - y_1 \) calculates the vertical change, while the denominator \( x_2 - x_1 \) calculates the horizontal change between the points. If the slope is positive, the line ascends to the right; if negative, it descends. A zero slope means the line is horizontal, and an undefined slope refers to a vertical line.

When calculating slope using the formula, keep in mind that simplification of algebraic fractions may be necessary to get the final answer in simplest form, as illustrated in the example problem.

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface defined by two axes: the horizontal x-axis and the vertical y-axis. These axes intersect at a point named the origin, which has coordinates \( (0, 0) \).Each point on the plane is defined by an ordered pair \( (x, y) \),indicating its position relative to the origin. Horizontal lines have equal y-coordinates, while vertical lines have equal x-coordinates. Understanding the coordinate system is essential when working with linear equations and finding the slope of a line.

Since each point has two coordinates, when working with the slope formula, it's important to always start with the y-coordinate (up and down movement) followed by the x-coordinate (left and right movement).

Algebraic Fractions

Algebraic fractions are fractions where the numerator, denominator, or both contain algebraic expressions. Similar to numerical fractions, algebraic fractions are simplified by finding common denominators, cancelling out like terms, or factoring and reducing expressions.

When dealing with slope calculations, you may encounter algebraic fractions during the process. It's important to be comfortable manipulating these fractions to simplify the slope to its lowest terms, ensuring a clear, concise representation of the rate of change of the line. The exercise provided is a practical example where algebraic fractions must be simplified to arrive at the final slope value.

Linear Equations

Linear equations describe straight lines and are presented in various forms such as slope-intercept (\( y = mx + b \)), point-slope (\( y - y_1 = m(x - x_1) \)), and standard (\( Ax + By = C \)) form. Here, 'm' represents the slope, and 'b' is the y-intercept - the point where the line crosses the y-axis.

The slope is a key component of a linear equation as it dictates the angle and direction of the line. A deep understanding of the slope and how to calculate it is essential for interpreting linear equations and graphing lines accurately. In the context of homework exercises, mastery of slope can help in understanding the behavior of linear relationships across various disciplines such as physics, economics, and biology.