Question

Plot each pair of points and determine the slope of the line containing the points. Graph the line. $$ (-1,1) ;(2,3) $$

Short answer

Slope: \(\frac{2}{3}\). Plot \((-1,1)\) and \((2,3)\); draw a line through them.

Step-by-step solution

Plot the Points

Plot the points \((-1,1)\) and \((2,3)\) on a coordinate plane. The point \((-1,1)\) is located one unit up from the origin and one unit to the left, while the point \((2,3)\) is three units up from the origin and two units to the right.

Calculate the Slope

Use the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \). For the points \((-1,1)\) and \((2,3)\), let \( (x_1, y_1) = (-1,1) \) and \( (x_2, y_2) = (2,3) \). Therefore, the slope \( m \) is calculated as: \[ m = \frac{3 - 1}{2 - (-1)} = \frac{2}{3} \]

Graph the Line

To graph the line, start at one of the points, for example, \((-1,1)\). Use the slope \(\frac{2}{3}\) to determine the rise over run. From \((-1,1)\), move up 2 units and right 3 units to reach another point on the line. Connect these points with a straight line.

Key concepts

Coordinate Plane

The coordinate plane is a two-dimensional surface where we can plot points, lines, and curves. It consists of two perpendicular lines called axes: the horizontal axis is the x-axis, and the vertical axis is the y-axis.

Each point on the coordinate plane is represented by an ordered pair \((x, y)\), where x is the horizontal distance from the origin (0,0), and y is the vertical distance.

For instance, to locate the point \((-1,1)\), move 1 unit left along the x-axis and 1 unit up along the y-axis. For the point \(2,3)\), move 2 units right and 3 units up. This system is essential for graphing and analyzing mathematical functions and relationships.

Slope Formula

The slope of a line measures its steepness or incline and is denoted by the letter m. The slope is calculated using two points on the line. The formula is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Here, \( (x_1, y_1)\) and \( (x_2, y_2)\) are coordinates of the two points.

Let's use the points \((-1,1)\) and \(2,3)\):

• Identify the coordinates: \( (x_1, y_1) = (-1, 1)\) and \( (x_2, y_2) = (2, 3)\)
• Substitute into the formula: \[ m = \frac{3 - 1}{2 - (-1)} = \frac{2}{3} \]

The result, \(\frac{2}{3}\), tells us that for every 3 units we move horizontally, the line rises 2 units. This is called the rise over run.

Understanding the slope helps in graphing the line and analyzing its behavior.

Graphing Lines

Graphing a line involves plotting points and connecting them to form the line. Once we have the slope and points:

1. Start at the given point, for example, \((-1,1)\).
2. Use the slope to find another point. For \( m = \frac{2}{3} \), move up 2 units and right 3 units from \((-1,1)\) to get to a new point, say, \( (2, 3) \).

After plotting the points, draw a straight line through them. This line represents the linear equation that passes through both points.

Graphing lines is crucial for visualizing relationships in algebra and geometry. It shows how variables interact and change together, making abstract concepts more concrete and understandable.