Problem 72 - Algebra A Combined Function

Question

Solve. See a Concept Check in this section. Given the points (2,3) and \((-5,1),\) can the slope of the line through these points be calculated by \(\frac{1-3}{2-(-5)} ?\) Why or why not? (See the second Concept Check.)

Step-by-Step Solution

Answer

No, the given expression does not properly apply the formula and results in an incorrect slope value.

Step 1: Recall the Slope Formula

The slope of a line (m) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Here, the changes in the vertical (y-values) and horizontal (x-values) directions are calculated.

Step 2: Identify the Coordinates

Identify the coordinates given in the problem:First point: \( (2, 3) \)Second point: \( (-5, 1) \)

Step 3: Apply the Slope Formula

Substitute the coordinates into the slope formula: Choose \( (x_1, y_1) = (2, 3) \) and \( (x_2, y_2) = (-5, 1) \) which gives:\[ m = \frac{1 - 3}{-5 - 2} \] Calculate the differences in the numerator and denominator.

Step 4: Calculate the Numerator and Denominator

Calculate the numerator: \\[ 1 - 3 = -2 \] Calculate the denominator: \[ -5 - 2 = -7 \]

Step 5: Calculate the Slope

Substitute the calculated values:\[ m = \frac{-2}{-7} \] Simplifying this gives the slope:\[ m = \frac{2}{7} \]

Step 6: Verify Calculation Method

Compare the original calculation question: Is \( \frac{1-3}{2-(-5)} \) valid?Breaking down:\[ \frac{1-3}{2+5} \] \[ \frac{-2}{7} ot= \frac{2}{7} \]The calculation differs from the correct value, meaning the arrangement of numerator or denominator is incorrect.

Key Concepts

Slope of a Line

The slope of a line is a value that describes how steep a line is on a graph. It tells us how much the line ascends or descends as it moves from left to right. The slope is crucial in linear relationships and is often denoted by the letter "m." To find the slope between two points, you use the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Let's take two arbitrary points on a graph:

• Point 1: \((x_1, y_1)\)
• Point 2: \((x_2, y_2)\)

In the context of the provided exercise, the points are \((2, 3)\) and \((-5, 1)\).
By substituting these into the slope formula: \[ m = \frac{1 - 3}{-5 - 2} = \frac{-2}{-7} \]Simplifying this fraction gives:\[ m = \frac{2}{7} \] This tells us for every 7 units the line moves to the right, it moves 2 units upwards. Thus, indicating the line is gently sloping upwards.

Coordinate Geometry

Coordinates in geometry consist of pairs of numbers that define a location on the Cartesian plane. The coordinate plane is divided into four quadrants, which help us locate the position of points. Each point is defined by an \((x, y)\) pair, where "x" represents the horizontal position and "y" the vertical position.In the example of the points provided:

• First Point: \((2, 3)\) indicates a position 2 units to the right and 3 units up from the origin (0,0).
• Second Point: \((-5, 1)\) is located 5 units left and 1 unit up from the origin.

These coordinates help in determining the line's path on the graph. By understanding their positions, we can accurately apply the slope formula to find the line's incline between any two points.

Mathematical Formulas

Mathematical formulas are like recipes in a kitchen; they offer a method to achieve a desired result. In algebra and specifically when discussing lines, the slope formula is vital. It's used to find the steepness of a line and is written as: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]Here, the difference in the "y-values" (\(y_2 - y_1\)) is your change in the vertical direction, while \((x_2 - x_1)\) is the change in the horizontal direction.
This formula ensures we have a consistent method for calculating slope, critical for graphing, understanding the direction of lines, and even in real-life applications like engineering or physics. Using formulas correctly ensures accuracy. In our exercise, inaccurate substitution like \[\frac{1-3}{2-(-5)}\] equates to \[\frac{1-3}{2+5} = \frac{-2}{7}\] This shows that switching the numerator's or denominator's order can lead to an incorrect slope calculation, illustrating the importance of careful application of formulas.