Problem 3 - Algebra 2

Question

Find the slope of the line that passes through each pair of points. $$ (4,5),(-1,0) $$

Step-by-Step Solution

Answer

The slope of the line is 1.

Step 1: Identify the formula

To find the slope of a line through two points, we use the slope formula:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points.

Step 2: Assign the points to the formula

Assign \((x_1, y_1)\) as \((4,5)\) and \((x_2, y_2)\) as \((-1,0)\). Substitute these into the formula:\[ m = \frac{0 - 5}{-1 - 4} \].

Step 3: Calculate the differences in coordinates

Calculate the differences:\( y_2 - y_1 = 0 - 5 = -5 \) and \( x_2 - x_1 = -1 - 4 = -5 \).

Step 4: Compute the slope

Substitute the differences into the formula:\[ m = \frac{-5}{-5} = 1 \]. Thus, the slope of the line is 1.

Key Concepts

Slope Formula

One crucial aspect of understanding the slope of a line in mathematics is the slope formula. This formula essentially helps us quantify the steepness of a line passing through two points in a plane. It is represented as:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Here,

• \((x_1, y_1)\) and \((x_2, y_2)\) denote the coordinates of two points on the line.
• The numerator \(y_2 - y_1\) calculates the vertical change (rise) between the points.
• The denominator \(x_2 - x_1\) calculates the horizontal change (run) between the points.

Using the slope formula, you can determine whether a line is horizontal (slope = 0), vertical (undefined slope), or has a positive or negative slope. Understanding this formula is essential for solving problems related to slope and interpreting the direction and angle of a line.

Coordinate Geometry

The field of coordinate geometry, also known as analytic geometry, is an important branch of mathematics. It serves as the bridge between algebra and geometry. Through coordinate geometry, mathematicians can study shapes and figures using an algebraic approach by placing them on a coordinate plane:

• In a coordinate plane, every point is represented by an ordered pair of numbers (x, y), where \(x\) is the horizontal location and \(y\) is the vertical.
• The use of coordinates allows us to perform algebraic manipulations on geometric figures to obtain formulas, such as the slope formula, to describe geometric properties of lines and curves.

By leveraging these coordinate systems, you can relate algebraic equations to geometric figures, making it easier to solve complex geometric problems. Coordinate geometry is a foundational tool for understanding more advanced topics in both algebra and geometry.

Linear Equations

Linear equations form the backbone of many mathematical concepts and practical applications. A linear equation is an equation that makes a straight line when it is graphed on a coordinate plane. Generally, it is written in the form:\[ y = mx + b \]where:

• \(m\) is the slope of the line, indicating its steepness.
• \(b\) is the y-intercept, the point where the line crosses the y-axis.

Linear equations allow us to model relationships between variables, such as predicting trends or analyzing data lines. They help in understanding how a change in one variable affects another. Furthermore, by using the slope-intercept form, we can easily graph a line by identifying both its slope and intercept, making it a handy tool in both mathematics and real-world scenarios.