Problem 14 - Algebra 1

Question

\(x=0\) a. (0,14) b. (14,0)

Step-by-Step Solution

Answer

The new sets of coordinates after substitution are: a. (0,14) b. (0,0)

Step 1: Substitute \(x=0\) in set a

For set a, we have (0,14). Since \(x=0\), we just replace the first value of the pair with 0, so the resulting pair is (0,14)

Step 2: Substitute \(x=0\) in set b

For set b, we have (14,0). Now, replace the value of \(x\) in this set which turns the pair into (0,0)

Key Concepts

Ordered Pairs

Ordered pairs are a fundamental concept in coordinate systems. They consist of two numbers written in a specific order within parentheses, like this:

• (x, y) where 'x' is the first element, defining the position along the horizontal axis (the x-axis), and 'y' is the second element, defining the position along the vertical axis (the y-axis).

Ordered pairs are crucial because they provide a way to describe a precise location on a plane. In contexts like geometry and algebra, these pairs help in locating points and graphing equations.
For example:

• In the ordered pair (0, 14), the number 0 is known as the x-coordinate and indicates the position along the x-axis, while 14 is the y-coordinate and represents the location along the y-axis.

Understanding ordered pairs is essential for solving many problems in mathematics involving graphs or interpreting coordinates.

Substitution Method

The substitution method is a problem-solving strategy often used in algebra, particularly for solving systems of equations. It involves substituting one variable's value with another variable or expression to simplify and solve equations.

• This method becomes especially useful when you have an equation and are given or know the value of one of the variables.

Let's look at the example exercise:

• Given the ordered pair (0, 14), if you are asked to check or verify if it's a solution to an equation where the x-value is known to be 0, you substitute 0 in place of x and check the results. Since it's already aligned with our solution (0, 14), it's a valid pair.
• Similarly, for checking how an ordered pair behaves when substituted, the second pair (14, 0) would, after substituting x with 0, become (0, 0), illustrating how substitution shifts the values.

Using substitution simplifies complex problems by reducing variables and allows for straightforward calculations with clear results.

Origin in Cartesian Plane

The origin is a critical point in any Cartesian coordinate system. It is represented as the ordered pair (0, 0) and serves as the reference point for the x and y axes.

• In the Cartesian plane, the origin is where the horizontal and vertical axes intersect.
• It is considered the center of the plane and is used as a starting point for plotting points along the axes.
• The origin is important because it provides a fixed point from which all other points' distances and directions are measured.

In our example, when you replace the x in (14, 0) with 0, the pair turns into (0, 0), which is exactly at the origin. This shows how changing the coordinates affects the position of a point on the plane, distinguishing whether a point lies exactly at the center or elsewhere.
Understanding the significance of the origin helps in accurately plotting and interpreting positions in the plane.