Problem 43 - Algebra A Combined Function

Question

Complete the table of ordered pairs for each linear equation. See Examples 6 and 7. $$ x=-y+2 $$ $$ \begin{array}{|c|c|} \hline \boldsymbol{x} & \boldsymbol{y} \\ \hline 0 & \\ \hline & 0 \\ \hline-3 & \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The ordered pairs are (0, 2), (2, 0), (-3, -5).

Step 1: Substitute x = 0

First, let's substitute \( x = 0 \) into the equation \( x = -y + 2 \).\[ 0 = -y + 2 \] Simplifying, we find \( y = 2 \). So, the ordered pair is \( (0, 2) \).

Step 2: Substitute y = 0

Next, substitute \( y = 0 \) into the equation \( x = -y + 2 \).\[ x = -0 + 2 \] So, \( x = 2 \). The ordered pair is \( (2, 0) \).

Step 3: Substitute x = -3

Now, substitute \( x = -3 \) into the equation \( x = -y + 2 \).\[ -3 = -y + 2 \] Solving for \( y \), we get \( y = -5 \). This gives us the ordered pair \( (-3, -5) \).

Step 4: Complete the Table

From our calculations, we have the ordered pairs: \((0, 2)\), \((2, 0)\), and \((-3, -5)\). Fill in the table as follows: \(\begin{array}{|c|c|}\hline \boldsymbol{x} & \boldsymbol{y} \hline 0 & 2 \hline 2 & 0 \hline -3 & -5 \hline\end{array}\).

Key Concepts

Understanding Ordered Pairs

An ordered pair is a fundamental concept in mathematics, especially when dealing with linear equations. It consists of two components, usually written as \((x, y)\). The first element of this pair represents the value on the x-axis, while the second element represents the value on the y-axis. By plotting these points on a graph, you can visualize relationships and patterns such as lines in a linear equation.

Ordered pairs are crucial because they provide a simple way to express solutions to equations. They clearly show where a particular equation "lives" on the graph. For example, in the equation \(x = -y + 2\), substituting values for \(x\) or \(y\) produces ordered pairs like \((0, 2)\), \((2, 0)\), and \((-3, -5)\). These can be plotted to reveal the line that represents our equation.

• The order matters: \((x, y)\) is not the same as \((y, x)\).
• These pairs make it easy to see how one variable changes with respect to the other.
• They are used in almost all graph-related problems and provide insight into the nature of equations.

The Substitution Method

The substitution method is a strategy used to solve equations by replacing one variable with an expression obtained from another equation. In linear equations, it's particularly handy and keeps things neat and systematic.

Here's how it generally works: You start with one equation and solve it for one of the variables. You then take the expression that equals this variable and substitute it into another equation. This substitution effectively reduces the number of variables, making it easier to solve.

• In our problem, we did substitutions such as \(x = 0\) or \(y = 0\).
• This method helps simplify the equations, leading us straight to the ordered pairs.
• Substitution is useful because it turns two equations into one simpler equation.

By using the substitution method, you learn how to manipulate expressions and concentrate on just one variable at a time. It is a powerful tool, especially for systems of equations and linear equations of the type we're considering here.

Solving Equations Step-by-Step

When it comes to solving equations, such as linear ones, it’s essential to approach them step-by-step. This systematic process ensures that you don’t make errors or overlook important details.

To solve an equation like \(x = -y + 2\), you would follow a clear sequence:
1. **Identify the equation** - Understand what you are working with. Here, it is a linear equation in two variables.2. **Substitute values** - As shown in the solution, you can substitute values for one variable at a time, hence converting the equation into a simpler form.3. **Simplify** - Carry out necessary algebraic manipulations to simplify the expressions.4. **Solve for the variable** - Isolate the variable you’re solving for and find its value.5. **Form ordered pairs** - Once you have both \(x\) and \(y\), write them as ordered pairs like \((0, 2)\), \((2, 0)\), and \((-3, -5)\).

By following these steps, you ensure that you find correct solutions systematically. It’s not just about finding an answer but about understanding how to arrive at that answer efficiently and accurately. This process is foundational not only in math but also in logical problem-solving as a whole.