Question

1. let x be the number of cakes magdalena can make, and y be the batches of cookies she can make. write an equation that shows the possible combinations of x and y when using all 24 eggs.
2. graph the relationship between the number of cakes, x, and the batches of cookies, y.
3. dashiell says the equation of this line is y = 12 - 1.5x. do you agree? explain.

Explanation:

Step1: Assume egg - usage per item

Let's assume each cake uses 2 eggs and each batch of cookies uses 1 egg. The total number of eggs is 24. So the equation for the combination of cakes \(x\) and batches of cookies \(y\) is \(2x + y=24\).

Step2: Rewrite the equation for graphing

We can rewrite \(2x + y = 24\) in slope - intercept form \(y=-2x + 24\). To graph this line, we find the \(y\) - intercept (when \(x = 0\), \(y=24\)) and the \(x\) - intercept (when \(y = 0\), \(2x=24\), so \(x = 12\)). Then we can draw a straight line through these two points \((0,24)\) and \((12,0)\) on the \(x - y\) plane where \(x\geq0\) and \(y\geq0\) since the number of cakes and batches of cookies cannot be negative.

Step3: Check Dashiell's equation

Dashiell says \(y = 12-1.5x\). From our equation \(2x + y=24\) which is \(y=-2x + 24\), Dashiell's equation is incorrect. The slope and the \(y\) - intercept in Dashiell's equation do not match the equation based on the egg - usage relationship.

Answer:

1. \(2x + y=24\)
2. Graph a line with \(y\) - intercept at \((0,24)\) and \(x\) - intercept at \((12,0)\) in the first - quadrant (\(x\geq0,y\geq0\))
3. No. Our derived equation is \(y=-2x + 24\), which has a different slope and \(y\) - intercept than \(y = 12-1.5x\).