Problem 54 - Intermediate Algebra

Question

Use the given equation to complete each table. $$ y=2 x+15 $$ (table cant copy)

Step-by-Step Solution

Answer

Calculate \( y \) using the equation: plug in \( x \) and solve for \( y \), filling the table with the results.

Step 1: Understanding the Equation

First, recognize the type of equation you are working with. The equation given is a linear equation in the form of \( y = mx + c \), where \( m = 2 \) (the slope) and \( c = 15 \) (the y-intercept). This equation describes a straight line on a graph.

Step 2: Set Up the Table

Identify the input values (usually \( x \) values) you'll use to calculate the corresponding \( y \) values. Without the table, you would typically fill it with values like \( -2, -1, 0, 1, 2 \) to see a pattern in the results.

Step 3: Calculating \( y \) for Each \( x \)-Value

Substitute each \( x \)-value from the table into the equation \( y = 2x + 15 \). Calculate the resulting \( y \) value for each substitution.- For \( x = -2 \): \( y = 2(-2) + 15 = -4 + 15 = 11 \)- For \( x = -1 \): \( y = 2(-1) + 15 = -2 + 15 = 13 \)- For \( x = 0 \): \( y = 2(0) + 15 = 0 + 15 = 15 \)- For \( x = 1 \): \( y = 2(1) + 15 = 2 + 15 = 17 \)- For \( x = 2 \): \( y = 2(2) + 15 = 4 + 15 = 19 \)

Step 4: Completing the Table

Use the results from your calculations to fill in the table with the corresponding \( y \) values for each \( x \) value:| \( x \) | \( y \) ||-------|-------|| -2 | 11 || -1 | 13 || 0 | 15 || 1 | 17 || 2 | 19 |

Key Concepts

Slope-Intercept Form

The slope-intercept form is one of the most common ways to express a linear equation. The general form of a straight-line equation is represented as \( y = mx + c \). Here, \( m \) represents the slope of the line, while \( c \) is the y-intercept.
This form is particularly useful because it immediately gives information about the line's characteristics.

• **Slope (\( m \))**: This tells you how steep the line is. It represents the change in \( y \) over the change in \( x \). For example, a slope of 2 means that for every increase of 1 in \( x \), \( y \) increases by 2.
• **Y-intercept (\( c \))**: This is where the line crosses the y-axis. It gives the value of \( y \) when \( x \) is zero. In this example, a y-intercept of 15 means that the line crosses the y-axis at the point (0, 15).

Understanding the slope and y-intercept helps in sketching the graph and predicting the line's behavior with various \( x \) values.

Mathematical Tables

Mathematical tables are a practical tool for organizing and calculating values that conform to a specific equation. They allow you to substitute specific \( x \) values into a linear equation to determine the corresponding \( y \) values. This aids in visualizing the linear relationship between the variables.To fill in a table, follow these steps:

• **Choose \( x \) values**: Select a range of \( x \) values, including negative, positive, and zero values like -2, -1, 0, 1, and 2. These provide a clear view of the linear pattern.
• **Substitute into the Equation**: For each \( x \) value, plug it into the equation and solve for \( y \). For example, substituting \( x = -2 \) into \( y = 2x + 15 \) yields \( y = 11 \).
• **Record in the Table**: Write down each resulting \( y \) value alongside its corresponding \( x \) value, forming a clear and organized table.

Using a table helps in systematically calculating multiple points that can later be used for graphing.

Graphing Linear Equations

Graphing linear equations is a method that visually represents the relationship between the variables \( x \) and \( y \) on a coordinate plane. This visual representation is key to understanding the equation's behavior and to see patterns or trends, like the linear nature of the relationship.Here’s how to proceed:

• **Plot Points**: Once you've used a table to calculate several \( x, y \) pairs, plot each point on the graph by locating the corresponding position in the coordinate plane.
• **Draw the Line**: Connect the plotted dots with a straight line. Make sure the line extends in both directions beyond the plotted points, showing that the line continues infinitely in both directions.
• **Check Features**: Ensure your drawn line crosses the y-axis at the y-intercept (in our case, at 15). Confirm the slope by checking how steep your line is compared to the axes.

Graphing gives a clear picture of how \( y \) changes with \( x \) and allows for easier prediction of the equation's output for any \( x \). It's an essential skill for visual learners and adds a geometric understanding to algebraic expressions.