Question

Find the slope of the line joining the points \((7,5)\) and \((9,7):\) (a) 1 (b) 2 (c) \(\frac{1}{2}\) (d) 3

Short answer

Answer: (a) 1

Step-by-step solution

Substitute Coordinates

We have two points \((x_1, y_1) = (7, 5)\) and \((x_2, y_2) = (9, 7)\). We will plug these values into the formula for slope.

Calculate Slope

Using the coordinates, we can calculate the slope. $$ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{7 - 5}{9 - 7} = \frac{2}{2} $$

Simplify Slope

Now, simplify the fraction to find the slope. $$ m = \frac{2}{2} = 1 $$ Thus, the correct answer is (a) 1, as the slope of the line joining the given points is 1.

Key concepts

Coordinate Geometry

In coordinate geometry, we focus on representing and analyzing geometric figures using a coordinate plane. It consists of two perpendicular axes: the x-axis (horizontal) and y-axis (vertical). Together, these axes form a grid where every point is defined by a pair of numerical coordinates
The key features of a coordinate plane include:

• **Points**: Each point is defined by an \(x, y\) coordinate, which denotes its position.
• **Quadrants**: The plane is divided into four sections known as quadrants.
• **Equation of a Line**: A line can be represented by an equation in this system using known formulas.

Coordinates make it easy to calculate distances, midpoints, and slopes, such as in the provided exercise where points \(7,5\) and \(9,7\) were used to find the line's slope.

Linear Equations

Linear equations are expressions that form straight lines when graphed on a coordinate plane. A common form of linear equations is the slope-intercept form: \(y = mx + b\). Here:

• **m** is the slope.
• **b** is the y-intercept, where the line crosses the y-axis.

For example, in our exercise, the slope \(m\) was calculated as 1. This means for every unit increase in x, y increases by 1 as well. Linear equations are foundational in graphing because they describe relationships with constant rates of change, thus creating straight lines on the graph.
The slope can be positive, negative, zero, or undefined, each describing a different type of line:

• **Positive slope**: Line rises from left to right.
• **Negative slope**: Line falls from left to right.
• **Zero slope**: Horizontal line.
• **Undefined slope**: Vertical line.

Understanding these concepts helps in identifying and graphing lines more effectively.

Graphical Analysis

Graphical analysis involves creating and interpreting graphs to visualize relationships between variables. When analyzing graphs of linear equations, the slope can provide insights into the behavior of the line:

• **Steepness**: A larger slope indicates a steeper line.
• **Direction**: Positive slopes go upward, negative slopes downward.

In our example, we found the slope of the line between points \(7,5\) and \(9,7\) to be 1. Graphically, this tells us the line makes a perfect 45-degree angle with both axes, indicating equal rise and run.
Analyzing graphs also entails understanding intercepts:

• **x-intercept**: The point where the line crosses the x-axis.
• **y-intercept**: The point where the line crosses the y-axis.

Using graphs enables quick visualization of solutions and understanding of relationships between variables, making them a powerful tool in both mathematics and real-world applications.