Problem 77 - Elementary Algebra

Question

For the following problems, find the slope of the line through the pairs of points. Round to two decimal places. $$ (5.56,9.37),(2.16,4.90) $$

Step-by-Step Solution

Answer

Answer: The approximate slope of the line is 1.32.

Step 1: Identify the coordinates of the given points

The problem gives us two points: (5.56, 9.37) and (2.16, 4.90). We will label the coordinates as follows: (x1, y1) = (5.56, 9.37) (x2, y2) = (2.16, 4.90)

Step 2: Use the slope formula

Now we will plug the coordinates into the slope formula, $$m = \frac{y2 - y1}{x2 - x1}$$. So, $$m = \frac{4.90 - 9.37}{2.16 - 5.56}$$.

Step 3: Calculate the slope

Perform the calculations in the numerator and denominator: $$m = \frac{-4.47}{-3.4}$$ Now, divide -4.47 by -3.4 to get the slope: $$m \approx 1.32$$. So, the slope of the line that passes through the given pairs of points is approximately 1.32 (rounded to two decimal places).

Key Concepts

Understanding Coordinates

In mathematics, coordinates are used to specify the position of points in space. Each point has an associated pair of numbers called coordinates. For example, the point (5.56, 9.37) has coordinates where 5.56 is the x-coordinate and 9.37 is the y-coordinate. These numbers come together to give a unique position on a 2-dimensional plane.

When dealing with coordinates, it is important to know how they are set up:

• x-coordinate: This is the horizontal position of the point on the graph.
• y-coordinate: This is the vertical position of the point on the graph.

In our example, point (5.56, 9.37) is one point and point (2.16, 4.90) is another. Knowing how to locate these on a grid is the first step in understanding how these points relate to each other.

Mastering the Slope Formula

The slope of a line measures its steepness and direction. It's a crucial concept in algebra and geometry. The slope formula is the tool that helps us calculate this property between any two points on a plane.

To find the slope (m), use the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Here's a quick rundown:

• \(y_2\) and \(y_1\): These are the y-coordinates of the two points.
• \(x_2\) and \(x_1\): These are the x-coordinates of the two points.

By plugging in our coordinates from the points (5.56, 9.37) and (2.16, 4.90) we calculate:\[ m = \frac{4.90 - 9.37}{2.16 - 5.56} \] This step is essential because it sets us up for the final calculation. Solving continues by finding the difference in both y-coordinates and x-coordinates.

Performing Numerical Calculations

Once the slope formula is set up with our coordinates, we switch gears to some straightforward number crunching.Subtract the y-coordinates (9.37 and 4.90):
\( 4.90 - 9.37 = -4.47 \) Next, subtract the x-coordinates (5.56 and 2.16):
\( 2.16 - 5.56 = -3.4 \) With these values, divide the results to get the slope:

• \( m = \frac{-4.47}{-3.4} \)

Both numerator and denominator are negative, and dividing them gives a positive result, as dividing two negative numbers results in a positive number.

Finally, approximate the division result:
\( m \approx 1.32 \). Rounding to two decimal places is necessary for clarity and precision, especially in different contexts where precise measurements of slope are necessary.