Systems of Equations Master

Review: Solving 2x2 Algebraically

Before jumping to 3 unknowns, let's remember how to solve 2 equations algebraically. This method (Elimination) will be vital for Merit/Excellence later.

Context: 2 Apples and 3 Bananas cost \$8. 5 Apples and 2 Bananas cost \$9.

Let $a$ = cost of an Apple. Let $b$ = cost of a Banana.

Step 1: Formulate

(1) $a$ + $b$ =

(2) $a$ + $b$ =

Incorrect. Attempt /3.

Solution

Equation 1: 2 Apples + 3 Bananas = \$8 $\rightarrow 2a + 3b = 8$

Equation 2: 5 Apples + 2 Bananas = \$9 $\rightarrow 5a + 2b = 9$

Setup Verified

Step 2: Eliminate $b$

Multiply Eq(1) by 2, and Eq(2) by 3 so both have $6b$.

Eq(1)×2: 4a + 6b = 16

Eq(2)×3: 15a + 6b = 27

Subtract Eq(1)×2 from Eq(2)×3:

$a$ =

Incorrect. Attempt /3.

Solution

$15a - 4a = 11a$

$27 - 16 = 11$

Result: $11a = 11$

Elimination Verified

Step 3: Solve for $a$ and $b$

If $11a = 11$, then $a = 1$. Substitute $a=1$ back into Eq(1): $2(1) + 3b = 8$.

$b =$

Incorrect. Attempt /3.

Solution

$2(1) + 3b = 8 \rightarrow 2 + 3b = 8 \rightarrow 3b = 6 \rightarrow b = 2$

Great! Apples are \$1, Bananas are \$2. You are ready for 3x3!

Forming the Matrix

Sophia’s Smoothies manufactures three types of smoothies: Xero, Yum, and Zap.

Xero needs 12g of ingredient A, 4g of B, and 4g of C.
Yum needs 8g of ingredient A, 10g of B, and 2g of C.
Zap uses 6g of ingredient A, 4g of B, and 10g of C.

Find the number of smoothies that can be produced using 1180g of Ingredient A, 900g of Ingredient B, and 920g of Ingredient C.

Step 1: Define Variables

Before writing numbers, what are we trying to calculate? Type the exact names of the smoothies.

Let $x$ = Number of smoothies

Let $y$ = Number of smoothies

Let $z$ = Number of smoothies

Incorrect spelling. Attempt /3.

Solution

We are trying to find the quantity of each smoothie. The names are Xero, Yum, and Zap.

Variables Locked

Step 2: Build 3 Equations

Equation 1: Total Ingredient A

$x$ + $y$ + $z$ =

Equation 2: Total Ingredient B

$x$ + $y$ + $z$ =

Equation 3: Total Ingredient C

$x$ + $y$ + $z$ =

Incorrect. Check your rows and columns. Attempt /3.

Solution

Read the text vertically by ingredient:
Ing A: 12x + 8y + 6z = 1180
Ing B: 4x + 10y + 4z = 900
Ing C: 4x + 2y + 10z = 920

Matrix Secured!

Solving the System

We have our matrix. For Achieved, we jump straight to the Graphics Calculator. However, for Merit/Excellence later, you must know how to eliminate variables algebraically.

Phase 1: Algebraic Elimination Merit Skill Preview

(1) $12x + 8y + 6z = 1180$
(2) $4x + 10y + 4z = 900$
(3) $4x + 2y + 10z = 920$

To eliminate $x$, we multiply Eq(2) by 3, then subtract it from Eq(1).

$12x + 8y + 6z = 1180$
$-$ $(12x + 30y + 12z = 2700)$
-------------------------

Fill in the resulting coefficients for Eq A:

$y$ + $z$ =

Incorrect. Attempt /3.

Solution

$8y - 30y = -22y$
$6z - 12z = -6z$
$1180 - 2700 = -1520$

Correct! $-22y - 6z = -1520$. You would repeat this with Eq(3) to get a 2x2 system. But for now, let's use the calculator!

Phase 2: Graphics Calculator

Step 1

Go to MENU, select EQUATION.

Step 2

Press F1 (SIML), then F2 (3 Unknowns).

Step 3

Enter the matrix and press SOLVE (F1).

GC MATRIX VIEW

[ 12 8 6 | 1180 ]

[ 4 10 4 | 900 ]

[ 4 2 10 | 920 ]

What does the calculator output?

$x =$

$y =$

$z =$

Incorrect. Attempt /3.

Solution

If you enter the matrix exactly as shown above, your GC outputs: $x=30, y=50, z=70$.

GC Values Verified

ACHIEVED REQUIREMENT

To get the grade, you must interpret the numbers in a real-world sentence. Fill in the blanks:

"According to the amounts of ingredients available, Sophia's Smoothies can produce exactly Xero smoothies, Yum smoothies, and Zap smoothies with no wastage."

Incorrect mapping. Attempt /3.

Solution

$x=30$ (Xero), $y=50$ (Yum), $z=70$ (Zap).

Excellent! That is a complete "Achieved" level answer.

The Hockey Squad

Try applying the same logic to a brand new context.

To raise money, the hockey squad is selling three types of cupcakes: Chocolate ($c$), Strawberry ($s$), and Vanilla ($v$).

A dozen Chocolate requires 200g flour, 50g sugar, and 2 eggs.
A dozen Strawberry requires 250g flour, 30g sugar, and 3 eggs.
A dozen Vanilla requires 280g flour, 55g sugar, and 4 eggs.

Form the system of equations assuming they have exactly 5000g of flour, 1000g of sugar, and 60 eggs available.

Flour Equation

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

Sugar Equation

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

Eggs Equation

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

Incorrect. Attempt /3.

Solution

Flour amounts: 200c + 250s + 280v = 5000
Sugar amounts: 50c + 30s + 55v = 1000
Egg amounts: 2c + 3s + 4v = 60

Perfect!

You have successfully mastered 3x3 System Formulation.

(Note: The GC solution for this yields decimals. We will explore what this means in Week 2!)