Problem 102 - Introductory and Intermediate Algebra for College Students 4th

Question

Will help you prepare for the material covered in the next section. In each exercise, evaluate $$\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$$ for the given ordered pairs \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\). $$\left(x_{1}, y_{1}\right)=(4,-2) ;\left(x_{2}, y_{2}\right)=(6,-4)$$

Step-by-Step Solution

Answer

-1

Step 1: Identify the coordinates

Recognize the given pairs of coordinates. The first pair of coordinates is (x1, y1) which is (4,-2) and the second pair of coordinates is (x2, y2) which is (6,-4).

Step 2: Substitution to slope formula

Substitute the coordinate values into the slope formula \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\). This gives us the expression \(\frac{-4 - (-2)}{6-4} = \frac{-4 -(-2)}{2}\). Ensure you consider the negative sign correctly during substitution.

Step 3: Simplify the expression

\(\frac{-4 -(-2)}{2}\) simplifies to \(\frac{-4 + 2}{2}\) because two negatives make a positive. Further simplifying gives \(\frac{-2}{2}\).

Step 4: Final calculation

Divide -2 by 2 to get a final answer of -1.

Key Concepts

Ordered Pairs

Understanding ordered pairs is a fundamental aspect when working with the coordinate system. Ordered pairs are used to specify the position of points on a plane. They are given as \( (x, y) \) where \( x \) and \( y \) represent numbers on the horizontal (x-axis) and vertical (y-axis) axis, respectively.

Think of ordered pairs like an address for a location on a map. You need to know both the 'street' (x-axis) and the 'floor' (y-axis) to exactly determine where you are heading. In algebra, each ordered pair corresponds to one and only one point on the coordinate plane. When you're given two pairs as in the exercise, you can see them as two distinct addresses and the slope formula then helps to determine the steepness of the path between them.

• First coordinate (x-axis value)
• Second coordinate (y-axis value)

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface divided by a horizontal line (x-axis) and a vertical line (y-axis). The point where these axes intersect is called the origin, marked \( (0, 0) \).

Every point on the plane can be described by an ordered pair indicating its distance from the origin along the x-axis and y-axis. Negative numbers are used for directions opposite the positive axes. For instance, the point \( (4, -2) \) from the exercise is found by moving four units to the right of the origin (along the x-axis) and two units down (along the y-axis).

Important Features of the Coordinate Plane:

• Quadrants: the plane is divided into four quadrants
• Origins: the starting point at \( (0, 0) \)
• Axes: the 'streets' of our grid map

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operation symbols that represent mathematical relationships. For example, in the slope formula \( \frac{y_2 - y_1}{x_2 - x_1} \), we have an expression that calculates the slope (steepness) of the line connecting two points on the coordinate plane.

When dealing with algebraic expressions, it's crucial to perform operations according to the order of operations: parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right). Incorrectly applying operations can lead to wrong answers. Also, remember to watch for signs: subtracting a negative is the same as adding a positive, as seen in our exercise with \( (-4) - (-2) \) simplifying to \( \frac{-2}{2} \) and ultimately resulting in -1.

• Variables: represent unknown values
• Constants: fixed values
• Operators: symbols that designate operations