Write each English sentence as an equation in two variables. Then graph the equation. The y-value is four more than twice the x-value.
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Step 1: Start by translating the English sentence into a mathematical equation. The sentence states that 'The y-value is four more than twice the x-value.' This can be written as: .
Step 2: Identify the two variables in the equation. Here, is the independent variable (input), and is the dependent variable (output).
Step 3: To graph the equation, create a table of values. Choose a few values for (e.g., -2, -1, 0, 1, 2) and calculate the corresponding values using the equation .
Step 4: Plot the points from the table of values on a coordinate plane. For example, if , then , so plot the point (0, 4). Repeat this for all chosen values.
Step 5: Draw a straight line through the plotted points, as the equation represents a linear relationship. Label the graph with the equation for clarity.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Linear Equations
Linear equations are mathematical statements that express a relationship between two variables, typically in the form y = mx + b, where m is the slope and b is the y-intercept. Understanding linear equations is essential for translating verbal statements into mathematical form, as they represent straight lines when graphed.
Translating English sentences into equations involves identifying the mathematical relationships described in the text. This requires recognizing keywords and phrases that indicate operations, such as 'more than' or 'twice,' and converting them into algebraic expressions that accurately represent the relationships between the variables.
Graphing linear equations involves plotting points on a coordinate plane that satisfy the equation. By determining key values, such as the slope and y-intercept, one can draw a straight line that represents all solutions to the equation. This visual representation helps in understanding the relationship between the variables and analyzing their behavior.