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Sophia's New Recipe

Sophia's Smoothies is redesigning their flavours. They are dropping Ingredient A and introducing Ingredient E.

  • Xero ($x$) uses 4g of B, 4g of C, and 6g of E.
  • Yum ($y$) uses 10g of B, 2g of C, and 11g of E.
  • Zap ($z$) uses 4g of B, 10g of C, and 9g of E.

They have 900g of B, 920g of C, and 1360g of E.

Equation 1: Ingredient B
$x$ + $y$ + $z$ =
Equation 2: Ingredient C
$x$ + $y$ + $z$ =
Equation 3: Ingredient E
$x$ + $y$ + $z$ =
Incorrect. Check your columns. Attempt /3.

Solution

Eq 1: $4x + 10y + 4z = 900$
Eq 2: $4x + 2y + 10z = 920$
Eq 3: $6x + 11y + 9z = 1360$

Matrix Secured!

The Zero Paradox

If you enter this into a Casio GC, it either says "Infinite Solutions" or "Ma ERROR". To achieve Excellence, we must prove what is happening algebraically.
(1) $4x + 10y + 4z = 900$
(2) $4x + 2y + 10z = 920$
(3) $6x + 11y + 9z = 1360$

Step A: Subtract Eq(2) from Eq(1) to eliminate $x$.

$(4x+10y+4z = 900) - (4x+2y+10z = 920)$
$y$ + $z$ =
Incorrect. $10 - 2 = ?$ Attempt /3.

Solution

$8y - 6z = -20$

Step B: Eliminate $x$ from Eq(1) and Eq(3). Multiply Eq(1) by 1.5, then subtract Eq(3).

Eq(1)×1.5:   $6x + 15y + 6z = 1350$ Subtract Eq(3): $-(6x + 11y + 9z = 1360)$
$y$ + $z$ =
Incorrect. $15 - 11 = ?$ Attempt /3.

Solution

$4y - 3z = -10$

Step C: Compare the two results.

(Result A)   $8y - 6z = -20$
(Result B)   $4y - 3z = -10$

Notice that Result A is exactly double Result B. If you multiply Result B by 2 and subtract them, what do you get?

$=$
Incorrect. $8y - 8y = ?$ Attempt /3.

Solution

$0 = 0$

The system is DEPENDENT. Infinite Solutions!

The General Solution

Because there are infinite solutions, we must describe them algebraically. We do this by setting $z = z$, and writing $x$ and $y$ as formulas based on $z$. Excellence Skill

Phase 1: Solve for $y$

Take our simplified 2-variable equation: $4y - 3z = -10$. Rearrange it to make $y$ the subject.

$4y = $ $z - 10$
$y = $ $z - 2.5$
Incorrect. Move $-3z$ over, then divide by 4. Attempt /3.

Solution

$4y = 3z - 10 \Rightarrow y = 0.75z - 2.5$

Phase 2: Solve for $x$

Substitute our new $y$ equation back into Eq(1): $4x + 10y + 4z = 900$

$4x + 10(0.75z - 2.5) + 4z = 900$
$4x + 7.5z - 25 + 4z = 900$
$4x + 11.5z - 25 = 900$

Rearrange to make $x$ the subject:

$4x = $ $z + 925$
$x = $ $z + 231.25$
Incorrect. Move $+11.5z$ over, then divide by 4. Attempt /3.

Solution

$4x = -11.5z + 925 \Rightarrow x = -2.875z + 231.25$

General Solution: $(-2.875z + 231.25,\; 0.75z - 2.5,\; z)$

Geometry & Reality

We have the formula for the infinite possibilities. Now we must explain the geometry, and find one specific, real-world integer answer for Sophia.

The Geometry

When a system has infinite solutions, it means the third equation is just a blend (linear combination) of the first two. Visually, the three planes do not meet at a single dot. Instead, they intersect along an infinitely long path.

Fill in the blank:
"The system is dependent because the planes intersect along a single , like the spine of a book."
Incorrect. Think of the shape of the intersection. Attempt /3.

Solution

They intersect along a LINE.

The Real World Context

Sophia cannot make negative or fractional smoothies. We must find values of $z$ that make $x$ and $y$ positive whole numbers.

Let's test $z = 14$ using our General Solution.

$y = 0.75(14) - 2.5 =$
$x = -2.875(14) + 231.25 =$
Incorrect arithmetic. Attempt /3.

Solution

$y = 10.5 - 2.5 = 8$
$x = -40.25 + 231.25 = 191$

Excellence Achieved! A valid setup is 191 Xero, 8 Yum, and 14 Zap smoothies.