Final answer:
To write the equation, we use the slope-intercept form, y = mx + b. We find the slope using the given points and substitute it and one point into the equation to find the value of b. The equation representing the balance on the gift card is y = -3.15x + 26.
Step-by-step explanation:
To write an equation that represents the balance on the gift card, we can use the slope-intercept form of a linear equation, which is y = mx + b, where y represents the balance, x represents the number of coffees, m represents the rate of change, and b represents the initial balance.
From the given information, we have two points: (2, $19.70) and (4, $13.40). We can use these points to find the slope, which is the rate of change. The slope is calculated as m = (change in y)/(change in x). Substituting the coordinates, we get m = ($13.40 - $19.70)/(4 - 2). Simplifying, we have m = -$3.15.
Now, we can substitute the slope and one of the points into the slope-intercept form to find the value of b. Using the point (2, $19.70), we can write the equation as $19.70 = (-$3.15)(2) + b. Solving for b, we find that b = $26.
Therefore, the equation that represents the balance on the gift card is y = -$3.15x + $26. The slope of -3.15 indicates that the balance decreases by $3.15 for every additional coffee purchased, and the initial balance is $26.