In Exercises 47–50, write each English sentence as an equation in two variables. Then graph the equation. The y-value is the difference between four and twice the x-value.

Verified step by step guidance

1. Identify the variables: Let \( y \) be the y-value and \( x \) be the x-value.
2. Translate the English sentence into an equation: 'The y-value is the difference between four and twice the x-value' can be written as \( y = 4 - 2x \).
3. Recognize that this equation is in the form of a linear equation \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
4. Determine the slope and y-intercept: From \( y = 4 - 2x \), the slope \( m = -2 \) and the y-intercept \( b = 4 \).
5. Graph the equation: Start by plotting the y-intercept (0, 4) on the graph, then use the slope \( -2 \) (which means down 2 units for every 1 unit to the right) to plot another point, and draw the line through these points.

Key Concepts

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. In this context, the equation represents a straight line on a graph, illustrating how changes in one variable affect the other.

Translating English Sentences to Equations

Translating English sentences into mathematical equations involves identifying the relationships and operations described in the sentence. In this case, the phrase 'the y-value is the difference between four and twice the x-value' can be expressed mathematically as y = 4 - 2x, where the operations of subtraction and multiplication are clearly defined.

Graphing Linear Equations

Graphing linear equations involves plotting points on a coordinate plane that satisfy the equation. For the equation y = 4 - 2x, one can calculate y-values for various x-values, plot these points, and draw a straight line through them, visually representing the relationship between x and y.