Problem 54 Use the given equation to comple... [FREE SOLUTION]

Chapter 1: Problem 54

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

Short Answer

Expert verified

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

Step by step solution

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.

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.

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 $

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 |

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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鈥檚 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.

91影视

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

Short Answer

Step by step solution

Understanding the Equation

Set Up the Table

Calculating \( y \) for Each \( x \)-Value

Completing the Table

Key Concepts

Slope-Intercept Form

Mathematical Tables

Graphing Linear Equations

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