Problem 68 - Algebra A Combined Function

Question

Complete each ordered pair for the given equation. See Section \(3.1 .\) \(y=2 x-6 ;(2, \quad)\)

Step-by-Step Solution

Answer

The completed ordered pair is (2, -2).

Step 1: Identify the Equation

The given equation is \( y = 2x - 6 \). This equation represents a linear relationship between \( x \) and \( y \).

Step 2: Substitute the x-value

The ordered pair is \((2, \quad)\) which means \( x = 2 \). Substitute \( x = 2 \) into the equation: \[ y = 2(2) - 6 \]

Step 3: Solve for y

Simplify the expression by performing the multiplication and subtraction: \[ y = 4 - 6 \] This simplifies to \[ y = -2 \].

Step 4: Complete the Ordered Pair

Now that we have found \( y = -2 \), the complete ordered pair is \( (2, -2) \).

Key Concepts

Understanding Ordered Pairs

An ordered pair is a fundamental concept in algebra, representing a pair of values that correspond to one another in a specific order. In the context of a coordinate plane, an ordered pair is written as

• \((x, y)\)

This denotes the position of a point on a graph. The first value is the \( x \)-coordinate, and the second is the \( y \)-coordinate.
These coordinates tell you where to place a point on the horizontal and vertical axes. Ordered pairs are used to express the solutions of linear equations consistently.
In this linear equation exercise, we started with an equation and an incomplete ordered pair \( (2, \, \square) \).

• Here, we knew the \( x \)-value, but not the \( y \)-value.

Finding the \( y \)-value involves inserting this \( x \)-value into the given equation and solving for \( y \). Once complete, our ordered pair communicates the precise point where the relationship described by the equation holds true on a graph.

The Substitution Method

The substitution method is a useful technique often employed to solve equations, especially when dealing with systems of equations. It involves replacing a variable in one equation with an equivalent expression from another equation. However, in single equations, it's used to find unknown values.
In this particular exercise, the incomplete ordered pair was \((2, \square)\). We needed to find the missing \( y \)-value by substituting the given \( x \)-value into the equation

• We set \( x = 2 \)

Substituting \( x \) in the equation \( y = 2x - 6 \) gives us

• \( y = 2(2) - 6 \)

This method directly places the known variable into the equation, simplifying our ability to find the missing piece.
Using the substitution method, not just in simple equations, but also in larger systems, can streamline the process of identifying values, making it a staple strategy in math.

Solving Equations with Simplicity

Solving equations is about isolating the unknown variable to find its value. In linear equations, like the one in this exercise, this involves straightforward arithmetic operations.

• The key steps typically include substitution, multiplication, addition or subtraction, and simplification.

In our example, after substituting \( x = 2 \) into \( y = 2x - 6 \), we proceeded to solve for \( y \).

• First, multiply \( 2 \times 2 \) to get \( 4 \).
• Then, subtract \( 6 \) from \( 4 \) resulting in \( y = -2 \).

The simplicity of linear equations makes them a good starting point when learning to solve equations.
It's important, however, to carry out each step methodically and carefully to ensure accuracy, as even simple mistakes in calculations can lead to incorrect results. Moreover, practice is essential. The more you practice solving these equations, the quicker and more intuitive it becomes.
Recognizing patterns in operations, like consistently solving for the unknown, enhances understanding and efficiency in future mathematical problems.