Problem 27 - Algebra 1

Question

Find three different ordered pairs that are solutions of the equation. \(y=\frac{1}{2}(4-2 x)\)

Step-by-Step Solution

Answer

The three ordered pairs that are solutions of the equation are: (0, 2), (1, 1), and (2, 0)

Step 1: Arbitrary Selection of x and Calculation of y

Let's choose an arbitrary value for \(x\), say \(x = 0\). Substituting \(x = 0\) in \(y=\frac{1}{2}(4-2 x)\), we get \(y = 2\)

Step 2: Second Arbitrary Selection of x and Calculation of y

Choose another arbitrary value for \(x\), say \(x = 1\). Substituting these values in \(y=\frac{1}{2}(4-2 x)\), we get \(y = 1\)

Step 3: Third Arbitrary Selection of x and Calculation of y

For the last ordered pair, lets choose \(x = 2\). Once again substitute \(x = 2\) in the equation \(y=\frac{1}{2}(4-2 x)\), we get \(y = 0\)

Key Concepts

Ordered Pairs

In the world of mathematics, ordered pairs are a fundamental concept that you will encounter often, especially in linear algebra and graphing. An ordered pair is simply a way of documenting two values that show a particular relationship, often denoted as \((x, y)\). The first value, \(x\), typically represents the horizontal position on a coordinate plane, while \(y\) represents the vertical position. This specific order is important because changing the order of the values would indicate a different point.

When you find ordered pairs that are solutions to a given equation, like in our example equation \(y=\frac{1}{2}(4-2x)\), it means these pairs of \((x, y)\) make the equation true. To generate these pairs, choose values for \(x\) and calculate the corresponding \(y\). This gives you the coordinates that satisfy the equation, providing a point you can plot on a graph.

• Pick any number for \(x\).
• Substitute \(x\) into the equation to solve for \(y\).
• Document the \((x, y)\) pair as your solution.

This process helps you understand the relationship between the variables in the equation.

Substitution Method

The substitution method is a way to find solutions for linear equations by replacing one variable with an expression involving the other variable. It's an efficient technique in algebra that simplifies the problem by reducing the number of variables, making calculations more straightforward.

In our given task, the equation is \(y=\frac{1}{2}(4-2x)\). Here, we substituted different values for \(x\) to find the corresponding \(y\) values. By making arbitrary choices for \(x\), we used substitution to solve for \(y\). This process transformed our equation into a simpler arithmetic problem:

• Choose a specific value for \(x\).
• Replace \(x\) in the equation with the chosen value.
• Perform the arithmetic operations to solve for \(y\).
• The resulting \(y\) gives you an ordered pair \((x, y)\).

Using substitution like this is a practical method not only for graphing but also for solving systems of equations, where you have more than one equation involving the same variables.

Solving Equations

Solving equations involves finding values for the variables that make the equation true. In mathematics, particularly when dealing with linear equations, this often means manipulating the equation to isolate one of the variables on one side of the equation.

For the equation \(y=\frac{1}{2}(4-2x)\), solving it involved calculating different \(y\) values for arbitrary selections of \(x\). Linear equations like this typically form a straight line when graphed, and finding several \((x, y)\) pairs helps establish that line.

• Select values for \(x\), the independent variable.
• Use algebraic steps to compute the dependent variable \(y\).
• Each selection gives a pair that can be plotted to visualize the solution graphically.
• Solving equations is an iterative process of computation and verification.

The goal is to find all possible pairs that satisfy the equation, thereby illustrating the full set of solutions. Practice with more examples improves your fluency in moving between the equation form and its graphical representation.