Problem 48 - Precalculus Essentials

Question

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.

Step-by-Step Solution

Answer

The equivalent mathematical equation is \(y = 4 - 2x\) or \(y = -2x + 4\), depending on preferred format. The graph would be a straight line with y-intercept at 4 and slope of -2.

Step 1: Formulate the equation

The sentence 'The \(y\) -value is the difference between four and twice the \(x\) -value' can be translated into the equation \(y = 4 - 2x\). This is because 'the difference between four and twice the \(x\) -value' refers to subtracting two times the \(x\) -value from four.

Step 2: Rearrange the equation (Optional)

This step is optional, depending on how one finds it easier to graph the equation. Some prefer the format \(y = mx + c\), which clearly shows the slope \(m\) and the y-intercept \(c\). The equation can be rearranged to this format, i.e., \(y = -2x + 4\).

Step 3: Graph the equation

To graph the equation \(y = -2x + 4\), start by plotting the y-intercept (when \(x = 0, y = 4\)), then use the slope to find other points. Since the slope is -2, for each increment of 1 in \(x\), \(y\) will decrease by 2. Plotting these points and joining them with a straight line will give the graph of the equation.