Problem 19 - Algebra 1: Concepts and Skills

Question

Find three ordered pairs that are solutions of the equation. $$ x=\frac{1}{2} $$

Step-by-Step Solution

Answer

The three ordered pairs that are solutions of the equation \(x=\frac{1}{2}\) are: \(\left(\frac{1}{2}, -1\right)\), \(\left(\frac{1}{2}, 0\right)\), and \(\left(\frac{1}{2}, 1\right)\).

Step 1: Understanding the Equation

Look at the given equation, \(x=\frac{1}{2}\). The value of x is constant and equal to \(\frac{1}{2}\) for all points on the graph of this equation. The y-value isn't specified, which means it can be any real number.

Step 2: Choosing Y-Values

It's arbitrary which y-values to choose because it can be any real number. For instance, let's select y-values to be -1, 0, and 1.

Step 3: Forming Ordered Pairs

For each chosen y-value, formulate an ordered pair. Since x is always \(\frac{1}{2}\), the ordered pairs are \(\left(\frac{1}{2}, -1\right)\), \(\left(\frac{1}{2}, 0\right)\), and \(\left(\frac{1}{2}, 1\right)\).

Key Concepts

Constants in Equations

When we talk about constants in equations, we're referring to numbers that remain fixed in value. In the equation \(x = \frac{1}{2}\), \(x\) is a constant because it always equals \(\frac{1}{2}\), no matter what. Constants are crucial because they define specific, unchanging parts of an equation. In our case, though \(x\) is fixed, the equation tells us that \(y\) could be any value. This unique setup helps us understand what lines would look like on a graph. Whenever you encounter an equation with a constant like this, it simplifies part of the problem by giving you a piece of the puzzle you don't need to guess about.

Graphing Equations

Graphing an equation involves plotting points based on the relationship between variables. In the equation \(x = \frac{1}{2}\), the graph is a vertical line at \(x = \frac{1}{2}\). Why vertical? Because no matter what, \(x\) does not change, even if \(y\) does. To graph such an equation, follow these simple steps:

• Identify the fixed value for \(x\), which is \(\frac{1}{2}\).
• Select various \(y\) values, such as -1, 0, and 1, which can be any real number.
• Plot the points: \(\left(\frac{1}{2}, -1\right)\), \(\left(\frac{1}{2}, 0\right)\), and \(\left(\frac{1}{2}, 1\right)\).

The result is a vertical line running through these points on the graph, showing how \(x\) stays constant despite changes in \(y\). This simplicity makes vertical lines easy to draw and understand.

Solutions of Equations

Solutions in equations refer to the set of values that satisfy an equation. For our equation \(x = \frac{1}{2}\), every solution is where \(x\) equals \(\frac{1}{2}\) and \(y\) can be any number. This means there are infinite solutions because there are infinite possible values for \(y\).Steps to determine solutions:

• Start with the fixed value from the equation—in this instance, \(x = \frac{1}{2}\).
• Select any real number for \(y\) and pair it with the constant \(x\).
• Form ordered pairs, such as \((\frac{1}{2}, -1)\), \((\frac{1}{2}, 0)\), \((\frac{1}{2}, 1)\).

These ordered pairs represent specific solutions to the equation. Each one corresponds to a point on the graph. In this case, all such points lie along the line \(x = \frac{1}{2}\). Understanding how to find solutions helps in recognizing patterns and connecting algebra to visual graphs. Whether dealing with a simple line or a complex equation, finding solutions reveals the true nature of mathematical relationships.