Problem 31 - Algebra 1

Question

Plot the points and find the slope of the line passing through the points. $$(-1,-1),(-3,-6)$$

Step-by-Step Solution

Answer

The slope of the line passing through the points (-1,-1) and (-3,-6) is 2.5.

Step 1: Plotting the Points

Plot the points (-1,-1) and (-3, -6) on a graph. Do this by starting at the origin (0,0), then move to the right if the x-coordinate is positive, or to the left if it's negative. Afterwards, move upwards if the y-coordinate is positive and downwards if it's negative. These steps will give you the accurate positions of the points on the graph.

Step 2: Computing the Slope

The slope of a line passing through two points (x1,y1) and (x2, y2) can be found using the formula \(m = (y_2 - y_1) / (x_2 - x_1)\). Here, we have (x1,y1) = (-1,-1) and (x2,y2) = (-3,-6). By substituting these values into the formula, we have \(m = (-6 - [-1]) / (-3 - [-1]) = (-5) / (-2) = 2.5\)

Step 3: Drawing the Line

Draw a straight line that passes through the points. This line represents the line whose slope has just been calculated.

Key Concepts

Plotting Points on a Graph

Understanding how to plot points on a graph is critical when dealing with mathematical problems involving coordinate systems. To get started, you'll want to get comfortable with the Cartesian coordinate plane, which is divided into four quadrants by the x-axis (horizontal) and y-axis (vertical). Each point is defined by a pair of coordinates: the x-coordinate and the y-coordinate.

When plotting points like \( -1, -1 \) and \( -3, -6 \), each pair represents (x, y). Start at the origin, where \( x = 0 \) and \( y = 0 \). If the x-coordinate is negative, go left from the origin; if it's positive, go right. As for the y-coordinate, a negative value means moving down, while a positive one means going up. Once you've moved to the correct x and y, you can place a dot to represent your point. Doing this for both creates a visual representation of their location relative to each other and helps in visualizing the concept of slope which we'll discuss next.

In practice, this forms the foundation of graphing lines and understanding other graph-related concepts in advanced mathematics and science.

Slope Formula

The slope of a line is a number that reflects its steepness or incline and is an essential concept in algebra and calculus. It's commonly represented as 'm'. When you're given two points on a line, you can calculate the slope by utilizing the slope formula \( m = (y_2 - y_1) / (x_2 - x_1) \).

To break this down further, 'y2' and 'y1' represent the y-coordinates of the two points, while 'x2' and 'x1' are the x-coordinates. The denominator reflects the horizontal change or 'run', and the numerator represents the vertical change or 'rise'. The slope tells you how many units the line rises or falls for each unit of horizontal movement to the right.

For example, with points \( (-1, -1) \) and \( (-3, -6) \), the slope calculated using the formula will tell us how steep the line is that connects these points. In this case, we have a slope of 2.5, indicating the line rises 2.5 units for every unit it moves to the right.

Graphing Linear Equations

Graphing linear equations is a way to visually represent the relationship between two variables. Any line on a graph can be represented by an equation of the form \( y = mx + b \), where 'm' is the slope and 'b' is the y-intercept. This equation tells us the exact steepness of the line and where it crosses the y-axis.

To graph a linear equation, start by pinpointing the y-intercept on the graph. Then, use the slope to determine the rise over run and plot another point. Draw a line through these points, extending it in both directions, and you've graphed your linear equation.

When you have two points, like in our example with \( (-1, -1) \) and \( (-3, -6) \), after calculating the slope, you have part of what you need to write the equation of the line and graph it. If you further know the y-intercept or can solve for it using one of the points, you can then graph the line with full knowledge of its path on the coordinate plane.