Systems of Equations Master

The Hockey Squad Returns

Last week, the Hockey Squad made Chocolate ($c$), Strawberry ($s$), and Vanilla ($v$) cupcakes. They had 5000g of Flour and 60 Eggs.

Flour: $200c + 250s + 280v = 5000$
Eggs: $2c + 3s + 4v = 60$

THE TWIST:

To save money, the squad reduces the amount of sugar needed. Chocolate now needs 20g, Strawberry needs 25g, and Vanilla needs 28g. They still have exactly 1000g of sugar available.

Update the System

Flour Equation (Unchanged)

$200c + 250s + 280v = 5000$

NEW Sugar Equation

$c$ + $s$ + $v$ =

Eggs Equation (Unchanged)

$2c + 3s + 4v = 60$

Incorrect. Read the text carefully. Attempt /3.

Solution

The new sugar amounts are 20g, 25g, and 28g. Total is 1000g.
Equation: $20c + 25s + 28v = 1000$

Matrix Updated!

Let's see what happens when we try to solve this.

The Contradiction

If you type this new matrix into your Graphics Calculator, you will not get an answer.

Phase 1: The Calculator Error

What message does your Casio Graphics Calculator display when you press SOLVE?

Incorrect. Attempt /3.

Solution

The calculator will display a Math Error.

Correct! It says "Ma ERROR". To get Merit/Excellence, we must prove WHY algebraically.

Phase 2: Algebraic Proof MERIT SKILL

Let's look at the Flour and new Sugar equations. We want to eliminate $c$, $s$, and $v$ all at once.

(Flour) $200c + 250s + 280v = 5000$
(Sugar) $20c + 25s + 28v = 1000$

Step A: Multiply the Sugar equation by 10.

$c$ + $s$ + $v$ =

Incorrect. Multiply EVERY number by 10. Attempt /3.

Solution

$20\times10=200, 25\times10=250, 28\times10=280, 1000\times10=10000$

Step B: Subtract the new Sugar equation from the Flour equation.

$200c + 250s + 280v = 5000$
$-$ $(200c + 250s + 280v = 10000)$
-----------------------------

$0 =$

Incorrect. $5000 - 10000 = ?$ Attempt /3.

Solution

$5000 - 10000 = -5000$

ALGEBRAIC PROOF COMPLETE: $0 \neq -5000$

Visualizing the Impossible

We just proved that $0 = -5000$. This algebraic contradiction has a direct geometric meaning. Let's see how this works in 2D space before we look at our 3D cupcakes.

2D Space: Parallel Lines

In a 2D graph ($x$ and $y$), two equations with the same coefficient ratio but different constants form parallel lines. They never intersect, hence "no solution".
$y = 2x + 1$ and $y = 2x + 3$

3D Space: Parallel Planes

In 3D space ($x, y, z$), an equation forms a flat surface called a plane. If two equations have the same ratio but different constants, they form parallel planes. Even if a third plane slices through them, there is NO single point where all three meet.

Geometry & Reality

Let's look at the actual Hockey Squad equations. To make it fit nicely in the 3D viewer, we've simplified the flour and sugar constants to 500 and 1000. Look closely at how the Blue and Red planes behave.

The Hockey Squad Planes

Fill in the blank:

"The system is inconsistent because the Flour and Sugar equations represent two planes."

Incorrect. Look at the 3D graph above. Attempt /3.

Solution

Planes that never intersect are called PARALLEL.

The Real World Context

Finally, we must tell the Hockey Squad what this math actually means for their bake sale.

"Because the equations are inconsistent, there is possible solution. This means the squad cannot bake the cupcakes using all the ingredients perfectly; there will inevitably be left over."

Incorrect. Attempt /3.

Solution

There is NO possible solution. There will inevitably be WASTAGE.

Outstanding! Merit Level Achieved. You have successfully proved the inconsistency.