Problem 33 - Algebra 1

Question

Plot the points and find the slope of the line passing through the points. $$(2,2),(-3,5)$$

Step-by-Step Solution

Answer

The slope of the line passing through the points (2,2) and (-3,5) is -0.6.

Step 1: Plotting the Points

First, plot the points (2,2) and (-3,5) on a graph.

Step 2: Calculating the Slope

Use the slope formula \[\frac{y2 - y1}{x2 - x1}\] to find the slope of the line. Substitute (2,2) to \((x1, y1)\) and (-3,5) into \((x2, y2)\). This gives us \[\frac{5 - 2}{-3 - 2}\] which simplifies to \[\frac{3}{-5}\] or -0.6.

Step 3: Understanding the Slope

The slope of -0.6 indicates that for every unit increase along the x-axis, the line decreases by 0.6 units along the y-axis.

Key Concepts

Graphing Points

Graphing points on a plane is the first step in understanding how lines and shapes can be represented in geometry. Each point consists of an ordered pair \(x,y\), where \(x\) denotes the horizontal position, and \(y\) denotes the vertical position on the coordinate plane.

When graphing points like (2,2) and (-3,5), you start at the "origin" (0,0), which is the point where the horizontal x-axis and vertical y-axis intersect. To plot (2,2):

• Move 2 units to the right on the x-axis from the origin.
• Then, move 2 units up on the y-axis.

For the point (-3,5):

• Start at the origin, move 3 units to the left on the x-axis because of the negative sign.
• Move upwards 5 units on the y-axis.

By firmly plotting these points, you visualize the start and end coordinates that a line can pass through in a two-dimensional plane.

Slope Formula

The slope of a line is a measure of its steepness and direction. It describes how much the line rises or falls between two points. To calculate this, we use the slope formula: \[ m = \frac{y2 - y1}{x2 - x1} \] where \(m\) represents the slope itself.

For the points (2,2) as \(x1, y1\) and (-3,5) as \(x2, y2\):

• Plug in the values: \[ m = \frac{5 - 2}{-3 - 2} \]
• Simplify to get \[m = \frac{3}{-5} = -0.6 \]

The negative sign indicates a decline. This means that the line falls as you move from left to right along the x-axis. It's important to remember that the slope tells us how the y-value changes concerning a change in the x-value.

Coordinate Plane

The coordinate plane is a two-dimensional plane used to graphically represent points, lines, and curves. It's made up of two axes: the horizontal x-axis and the vertical y-axis, which intersect at the origin.

Each point on this plane is defined by an ordered pair (x, y). The x-coordinate shows the position as either left or right of the origin, while the y-coordinate specifies the position up or down. This standard framework helps in visualizing algebraic relationships.

• The plane is divided into four quadrants.
• Quadrant I is situated at the top-right where both coordinates are positive.
• Quadrant II is top-left with a negative x and a positive y.
• Quadrant III is bottom-left with both coordinates negative.
• Quadrant IV is bottom-right where the x is positive, and the y is negative.

Understanding this system is essential for plotting graphs effectively and can lay the foundation for exploring more complex mathematical concepts.