Problem 15 - 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. \((3,1)\) and \((-3,-2)\)

Step-by-Step Solution

Answer

The slope of the line that passes through the points (3,1) and (-3,-2) is 0.5.

Step 1: Plot the Points

Plotting the points \( (3,1) \) and \( (-3,-2) \) on the Cartesian plane.

Step 2: Draw Line

Once the points are plotted, draw a line that passes through both points on the graph.

Step 3: Determine Rise and Run

The rise is the vertical change between the points, calculated as \( 1 - (-2) = 3 \). The run is the horizontal change between the points, calculated as \( 3 - (-3) = 6 \).

Step 4: Compute Slope

The slope is given by the formula \( slope = rise / run \). Substituting the obtained values, \( slope = 3 / 6 = 0.5 \).

Key Concepts

Coordinate plane

The coordinate plane, also referred to as the Cartesian plane, is a two-dimensional surface where we can visually represent points, lines, and figures using an ordered pair of numbers. Each point is specified by a pair

• The horizontal axis is known as the \( x \)-axis.
• The vertical axis is known as the \( y \)-axis.

The point where these two axes intersect is called the origin, denoted as \((0,0)\). Each location on the plane can be represented by an \( x \) and \( y \) coordinate, representing the position relative to the origin.

When working with problems that involve plotting or finding the slope of a line, it's important to have a good understanding of these fundamental components of the coordinate plane. Visualizing the plane with this understanding will help clarify how points are depicted and lines are drawn across it.

Plotting points

Plotting points accurately on the coordinate plane is the first crucial step when dealing with lines and slopes. Each point is defined by its coordinates, typically written as \((x, y)\).

Here's how you can plot a point like \((3, 1)\):

• Start from the origin \((0,0)\).
• Move horizontally along the \(x\)-axis by the first number, 3 units in this case.
• From that position, move vertically by the second number, 1 unit.

Now, for the point \((-3, -2)\):

• Again, start from the origin \((0,0)\).
• Move horizontally by -3 units (left direction on the \(x\)-axis).
• Move down by -2 units on the \(y\)-axis.

Once both points are plotted, drawing a line through them connects them on the plane, setting the foundation to find their slope.

Finding slope

Understanding the slope of a line on the coordinate plane is critical for understanding how steep the line is. The slope \(m\) is a measure that describes the line's inclination, telling us how much the line goes up or down as it travels from left to right. To find the slope between two points, we use the formula:\[ m = \frac{\text{rise}}{\text{run}} \]The 'rise' is the vertical change, and the 'run' is the horizontal change between two points.

For the points \((3,1)\) and \((-3,-2)\):

• Calculate the rise: The \(y\)-coordinates change from -2 to 1, so \(1 - (-2) = 3\).
• Calculate the run: The \(x\)-coordinates change from -3 to 3, so \(3 - (-3) = 6\).

The slope \(m\) is then \(\frac{3}{6} = 0.5\).
This 0.5 indicates for every unit the line travels horizontally (run), it rises vertically (rise) by half a unit. Understanding these changes helps to easily predict and describe the line's behavior on the coordinate plane.