Problem 41 - Algebra A Combined Function

Question

Complete the table of ordered pairs for each linear equation. See Examples 6 and 7. $$ y=-7 x $$ $$ \begin{array}{|c|c|} \hline \boldsymbol{x} & \boldsymbol{y} \\ \hline 0 & \\ \hline-1 & \\ \hline & 2 \\ \hline & \\ \hline \end{array} $$

Step-by-Step Solution

Answer

Complete table: (0, 0), (-1, 7), (-\frac{2}{7}, 2), (1, -7).

Step 1: Understanding the Equation

The equation given is \( y = -7x \). This means that for any value of \( x \), the value of \( y \) is calculated by multiplying \( x \) by -7.

Step 2: Calculate y for x = 0

Using the equation \( y = -7x \), substitute \( x = 0 \) into the equation to find \( y \). Thus, \( y = -7 \times 0 = 0 \). Enter this value into the table.

Step 3: Calculate y for x = -1

Substitute \( x = -1 \) into the equation \( y = -7x \). So, \( y = -7 \times (-1) = 7 \). Enter this into the table.

Step 4: Calculate x for y = 2

Given \( y = 2 \), use the equation to solve for \( x \). Rearranging gives \( x = \frac{y}{-7} = \frac{2}{-7} = -\frac{2}{7} \). Enter this value into the table.

Step 5: Select a Value for x and Calculate y

Choose a value for \( x \), e.g., \( x = 1 \), and calculate \( y \) using the equation \( y = -7x \). When \( x = 1 \), \( y = -7 \times 1 = -7 \). Enter these values into the table.

Key Concepts

Ordered Pairs

In mathematics, ordered pairs are used to locate points on a graph. An ordered pair is written as \((x, y)\), where \(x\) is the horizontal coordinate, and \(y\) is the vertical coordinate.
This pair gives specific information about the position of a point on a two-dimensional plane.

For example, an ordered pair \((0, 0)\) indicates the origin, where both coordinates are zero. Similarly, \((-1, 7)\) shows a point one unit to the left of the origin and seven units up. Knowing how to interpret ordered pairs is essential in understanding relationships between variables in equations.

Graphing Linear Equations

Graphing linear equations involves plotting points on a coordinate plane and connecting them to show how variables are related.
In the case of the equation \(y = -7x\), the line represents all the solutions to the equation.

To graph, you start by calculating several ordered pairs from the equation. For instance:

• When \(x = 0\), \(y = -7 \times 0 = 0\) gives the point \((0, 0)\).
• When \(x = -1\), \(y = -7 \times (-1) = 7\) results in the point \((-1, 7)\).
• With \(y = 2\), solving for \(x\) gives \(x = -\frac{2}{7}\); thus, \((-\frac{2}{7}, 2)\).

Once these points are plotted, they can be connected with a straight line.This line visually represents the relationship given by the equation, showing how \(y\) changes as \(x\) changes. Each point on the line is a solution to the equation, demonstrating the continuous nature of linear relationships.

Solving Equations Step by Step

Solving equations step by step is crucial for clarity and accuracy, especially in linear equations like \(y = -7x\).
It ensures you know exactly how to find each variable.

Here's a general approach:

• Identify the Equation: Understand the relationship it defines. \(y = -7x\) shows \(y\) as a function of \(x\). Each operation performed reveals the structure of a linear equation.
• Isolate Variables: Use algebra to solve for unknowns. For example, if you want to find \(x\) when \(y = 2\), rearrange to \(x = \frac{y}{-7}\).
• Substitute Known Values: Input known values for \(x\) or \(y\) into the equation to find the corresponding variable. This is essential for completing ordered pairs.
• Verify Results: Always check your calculations to ensure accuracy and logic consistency.

By following these steps, you develop a systematic approach to problem-solving, making even complex equations manageable.