Problem 21 - Algebra 1

Question

Plot the points and find the slope of the line passing through the points. $$(1,5),(5,2)$$

Step-by-Step Solution

Answer

The slope of the line passing through the points (1,5) and (5,2) is -3/4.

Step 1: Plot the Points

Step 1 involves plotting the given points, (1,5) and (5,2), on a graph. Each pair represents the coordinates (x,y) on the graph where x is the point on the horizontal axis (abscissa) and y is the point on the vertical axis (ordinate). You will end up with two distinguishable dots on your paper or screen.

Step 2: Calculate the Slope

The slope of the line passing through these two points can be calculated using the formula: \( slope = \frac{y_2 - y_1}{x_2 - x_1} \), where \( (x_1,y_1) \) and \( (x_2,y_2) \) are the coordinates of the two points. Using the given points this results in: \( slope = \frac{2 - 5}{5 - 1} = \frac{-3} {4} \)

Key Concepts

Plotting Points

Understanding how to plot points is a fundamental skill in coordinate graphing. Imagine a graph as a map where every location has a specific address. For plotting points, the 'address' is given as an ordered pair of numbers, like (x, y). The first number, x, tells us how far to move horizontally, either to the right for positive numbers or to the left for negative numbers. The second number, y, indicates the vertical position, moving up for positive numbers and down for negative numbers. When you plot both points from the exercise, (1,5) and (5,2), they should appear as dots on your graph where those lines intersect.

Slope Formula

The slope of a line is a measure of its steepness and direction. It's like thinking of a hill: the steeper the hill, the larger the slope. If the hill goes up as you move forward, the slope is positive; if it goes down, the slope is negative. Mathematically, you can find the slope using a simple formula: \(slope = \frac{y_2 - y_1}{x_2 - x_1}\). The formula compares the vertical change (rise) to the horizontal change (run) between two points on a line. Applying this formula to our points, we can calculate the slope by subtracting the y-values and dividing by the difference of the x-values.

Coordinate Graphing

Coordinate graphing is like playing a game of treasure hunt where the map is a grid of horizontal and vertical lines. This grid is also known as the Cartesian coordinate system. Each point on this grid corresponds to an ordered pair (x, y). When plotting points on this system, you start from the center, called the origin (0,0), and count the spaces to get to the x (horizontal) and y (vertical) values of your points. Plotting the points given in our exercise (1,5) and (5,2), helps to visualize the line's slope and provides a graphical representation of algebraic concepts in action.

Algebraic Concepts

Algebraic concepts provide a foundation for understanding mathematical relationships and solving problems. These concepts encompass operations involving numbers and variables, and understanding how to manipulate and interpret them algebraically. In the context of our exercise, algebra is used to compute the slope of a line, which is a linear equation representing the relationship between two variables, x and y. By plotting points and calculating the slope, we're applying these algebraic principles to gain a clearer understanding of how changes in one variable affect another in a graphical way.