Problem 13 - Algebra 1: Concepts and Skills

Question

Plot the points and draw a line that passes through them. Use the rise and run to find the slope. $$ (2,3) \text { and }(0,6) $$

Step-by-Step Solution

Answer

The slope of the line passing through the points (2,3) and (0,6) is -1.5

Step 1: Plot the Points

Plot the two points (2,3) and (0,6) on a graph. Point (2,3) is plotted by moving 2 units along the x-axis and 3 units up along the y-axis from the origin. Point (0,6) is plotted by moving 0 units along the x-axis and 6 units up along the y-axis from the origin.

Step 2: Draw a Line Through the Points

Draw a line that passes through the two points. Use a ruler or other straightedge to draw it accurately.

Step 3: Calculate the Rise

To calculate the rise (change in y-coordinates), subtract the y-coordinate of the second point from the y-coordinate of the first point. The y-coordinate of the first point is 3, and of the second point is 6. So, the rise = 6 - 3 = 3.

Step 4: Calculate the Run

To calculate the run (change in x-coordinates), subtract the x-coordinate of the second point from the x-coordinate of the first point. The x-coordinate of the first point is 2, and of the second point is 0. So, the run = 0 - 2 = -2.

Step 5: Calculate the Slope

The slope of the line is calculated as the ratio of the rise to the run. That is, slope = rise/run = 3/(-2) = -1.5. So, the slope of the line that passes through the points (2,3) and (0,6) is -1.5.

Key Concepts

Understanding Rise and Run

The concepts of "rise" and "run" are crucial when discussing the slope of a line. The rise is the change in the y-coordinates between two points on a line, whereas the run is the change in the x-coordinates. To better understand, think of the "rise" as how much you move up or down, and "run" as how much you move left or right when going from one point to another on the coordinate plane.
To find these values:

• Identify the first point, in our case, (2,3), and the second point, (0,6).
• The rise is determined by subtracting the y-coordinates: 6 - 3 = 3. This means you go up 3 units.
• The run is found by subtracting the x-coordinates: 0 - 2 = -2. This indicates a movement of 2 units to the left.

Using these calculations, the slope of the line that joins these points is then the rise divided by the run, which is \(\frac{3}{-2} = -1.5\). A negative slope like this suggests the line decreases as you move from left to right.

Plotting Points

Plotting points on a graph is a fundamental skill in understanding and applying mathematical concepts like slope. Each point in a coordinate plane is represented by an ordered pair (x, y). These pairs tell you exactly where to place your points on a grid.

• For point (2,3), begin at the origin (0,0) of the graph. Move 2 units to the right for the x-coordinate, and 3 units up for the y-coordinate.
• For point (0,6), start again at the origin. Since the x-coordinate is 0, do not move horizontally. Then move 6 units up, vertically, to mark this point.

Plotting points accurately helps you visualize the relationship between them, especially when multiple points form a particular line or curve.

The Coordinate Plane

The coordinate plane is a two-dimensional surface where each location is defined by a pair of numbers, known as coordinates (x, y). This plane has two perpendicular lines: the horizontal x-axis and the vertical y-axis, which intersect at the origin (0,0).

• The x-axis allows you to determine horizontal positions; moving right increases x-values, moving left decreases them.
• The y-axis helps you measure vertical positions; moving up increases y-values, moving down decreases them.

Hence, each point is pinpointed based on how far it lies along the x and y axes. This system is extremely useful for plotting lines and shapes, solving equations, and visualizing mathematical concepts on a broader scale. Working with the coordinate plane gives you a clear visual aid to understand geometric and algebraic ideas in math.