Problem 103 - Algebra and Trigonometry Real Mathematics, Real People

Question

Plot the points and find the slope of the line passing through the points. $$(0,1),(2,5)$$

Step-by-Step Solution

Answer

The slope of the line passing through the points (0,1) and (2,5) is 2.

Step 1: Plotting the given points

Plot the points (0,1) and (2,5) on the graph. These two points indicate two locations on the x and y axes where the line will pass through.

Step 2: Finding the Slope

Use the slope formula m=(y2-y1)/(x2-x1). Here, (x1, y1) is (0, 1) and (x2, y2) is (2, 5). Substituting these into the formula, m=(5-1)/(2-0). Thus, m=4/2=2.

Step 3: Analyzing the Slope

It is found that the slope of the line passing through the points (0,1) and (2,5) is 2. This slope indicates that the line raises by 2 units on the y-axis for each unit it moves right on the x-axis.

Key Concepts

Plotting Points

Plotting points is an essential step when working with linear equations. Every point on a graph represents a pair of coordinates, denoted as \((x, y)\), where \(x\) is the position on the horizontal axis and \(y\) is the position on the vertical axis. To plot the points \((0,1)\) and \((2,5)\), start on the x-axis at 0 and move up to 1 on the y-axis to mark the first point. This begins our line at the y-intercept. Next, from the origin, count over to x=2 and up to y=5 to place the second point.

Plotting these points gives you a guideline for drawing your line. By visually analyzing the placement of the points, you can start to see the direction and steepness of the line that connects them. It's beneficial to practice plotting accurately,

• because it helps you understand the graph's slope and line direction more intuitively
• and ensures that all your calculations using these points will be correct

Slope Formula

Once you have the points plotted, you can calculate the slope of a line that passes through them using the slope formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\). This equation allows you to determine how much a line rises or falls as it moves to the right.

The slope, denoted by \(m\), is a measure of the line's steepness. For the points \((0,1)\) and \((2,5)\), apply the formula:

• Assign \((x_1, y_1) = (0, 1)\) and \((x_2, y_2) = (2, 5)\).
• Substitute these values into the formula: \(m = \frac{5-1}{2-0} = \frac{4}{2} = 2\).

Understanding the slope as 2 means that for every single unit increase in \(x\), the \(y\) value increases by 2 units. Analyzing this helps in predicting future points on the line without plotting every single one, and confirms how changes in horizontal position affect vertical position.

Graphing Linear Equations

Graphing linear equations involves a combination of plotting points and understanding the slope to draw the full line on a graph. After plotting the initial given points and figuring out the slope, you have the tools to extend the line across the graph.

This process involves:

• Drawing a straight edge or line through the plotted points.
• Using the slope to extend the line by going up or down and over equal spaces.

A line with a slope of 2 will move upwards steeply, meaning you go up 2 blocks on the y-axis for every 1 block you move right on the x-axis. During this stage, it can be helpful to:

• check consistency by using another pair of points on the line
• or verifying calculations by plugging additional points into the linear equation matching your line.

Mastering these skills in graphing provides you with a powerful visual tool to solve problems involving linear equations efficiently.