Problem 63 - Algebra 2

Question

Find the slope of the line that passes through each pair of points. $$ (2,5),(6,9) $$

Step-by-Step Solution

Answer

The slope of the line is 1.

Step 1: Understand the Slope Formula

The slope of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated as follows:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]where \(m\) represents the slope.

Step 2: Identify the Coordinates

For the points \((2, 5)\) and \((6, 9)\):- \(x_1 = 2\)- \(y_1 = 5\)- \(x_2 = 6\)- \(y_2 = 9\).

Step 3: Apply the Slope Formula

Substitute the identified coordinates into the slope formula:\[m = \frac{9 - 5}{6 - 2}\]

Step 4: Calculate Numerator and Denominator

Calculate the difference in y-coordinates: \(9 - 5 = 4\).Calculate the difference in x-coordinates: \(6 - 2 = 4\).

Step 5: Compute the Slope

Complete the calculation by dividing the differences:\[m = \frac{4}{4} = 1\]So, the slope of the line is 1.

Key Concepts

Points on a Line

Understanding points on a line is a fundamental concept in geometry and algebra. When you have two points on a line, each point is represented by a pair of numbers known as coordinates. These coordinates are typically written as \(x, y\), where \(x\) indicates the position along the horizontal axis and \(y\) indicates the position along the vertical axis.

In our exercise, the points are \( (2, 5) \) and \( (6, 9) \). Here, the first number in each pair, such as 2 and 6, represents the x-coordinates. The second numbers, 5 and 9, represent the y-coordinates.

• The line passing through these two points will have specific properties such as direction and steepness, which can be determined using these coordinates.
• The slope of such a line gives us a measure of how steep or flat the line is between these two points.

Coordinate Geometry

Coordinate geometry involves using a coordinate plane and algebra to study geometric figures. The plane is divided into four quadrants by a horizontal line, called the x-axis, and a vertical line, known as the y-axis.

Each point in the plane can be pinpointed using an x and y coordinate. Coordinate geometry is immensely helpful in solving problems involving lines and shapes, as it combines spatial understanding with numeric calculations.

• It allows us to graphically represent algebraic equations.
• The relationship between points, lines, and slopes can be visualized on this plane, providing a deeper understanding of their properties.

Algebraic Formula

An algebraic formula is a set of rules expressed in symbols to calculate a specific value. In the case of calculating the slope of a line between two points, the specific formula used is:\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]This formula allows us to calculate the slope, denoted by \(m\), by finding the difference in the y-coordinates and dividing it by the difference in the x-coordinates of the two points. This helps determine how much the y-value of a line changes for a change in the x-value.

• By using algebraic formulas, we can systematically solve problems involving linear relationships.
• The formula ensures that calculations are accurate and consistent, making it a powerful tool when dealing with multiple data points.

Linear Equations

Linear equations are mathematical expressions that describe a line. They are represented in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. In our slope exercise, understanding the linear equation helps put the slope into context, as it determines the direction and steepness of the line.

By solving for the slope using two given points, you gain insight into the equation of that line, which can be further used to calculate other points on the same line.

• Linear equations are central to coordinate geometry as they allow us to predict the behavior of a line.
• Knowing the slope and one point on a line enables you to write the equation of the line and understand its trajectory in the coordinate plane.