Question

Question 1: Classify the system of equations and identify the number of solutions. 7x + 3y = 10 3y = 9 − 7x Answers: consistent, dependent; one inconsistent; one inconsistent; none consistent, dependent; infinite Question 2: Classify the system of equations and identify the number of solutions. 3x + y = 4 12x + 4y = 16 Answers: consistent, independent; one inconsistent; none consistent, dependent; one consistent, dependent; infinite Question 3: John and Harry went to a stationery shop. John bought 3 pens and 8 notebooks for $20.50. Harry bought 4 pens and 5 notebooks for $16.00. Identify the cost of a pen and the cost of a notebook. Answers: pen: $1.00; notebook: $2.50 pen: $2.00; notebook: $1.50 pen: $1.50; notebook: $2.00 pen: $2.00; notebook: $3.00

Answer 1

Answer:

1. The equations is inconsistent and has no solution

2. The equations is consistent and has no solution

3. A pen costs $1.50 and a notebook costs $2.00

Step-by-step explanation:

Solving $7x + 3y = 10$ and $3y = 9 - 7x$

$7x + 3y = 10$ --- Equation 1

$3y = 9 - 7x$ --- Equation 2

Add 7x to both sides in equation 2

$7x + 3y = 9 - 7x + 7x$

$7x + 3y = 9$  --- Equation 3

Subtract equation 3 from 1

$(7x + 3y = 10) - (7x + 3y = 9)$

$7x - 7x + 3y - 3y = 10 - 9$

$0 + 0 = 1$

$0 \neq 1$

The equations is inconsistent and has no solution

Solving $3x + y = 4$ and  $12x + 4y = 16$

$3x + y = 4$ -- Equation 1

$12x + 4y = 16$ ---- Equation 2

Make y the subject of formula in equation 1

$y = 4 - 3x$

Substitute 4 - 3x for y in equation 2

$12x + 4(4 - 3x) = 16$

$12x + 16 - 12x = 16$

Collect Like Terms

$12x - 12x = 16 - 16$

$0 = 0$

The equations is consistent and has no solution

3. Solving John and Harry

Given;

Represent Pen with P and Notes with N

John: $3P + 8N = 20.5$

Harry: $4P + 5N = 16.00$

Required

Find P and N

Make P the subject of formula in equation 2

$4P = 16 - 5N$

$P = \frac{16 - 5N}{4}$

Substitute $P = \frac{16 - 5N}{4}$ in equation 1

$3P + 8N = 20.5$

$3(\frac{16 - 5N}{4}) + 8N = 20.5$

Open the bracket

$\frac{48 - 15N}{4} + 8N = 20.5$

Take LCM

$\frac{48 - 15N + 32N}{4} = 20.5$

$\frac{48 + 17N}{4} = 20.5$

Multiply both sides by 4

$4 * \frac{48 + 17N}{4} = 20.5 * 4$

$48 + 17N = 82$

Subtract 48 from both sides

$48 - 48 + 17N = 82 - 48$

$17N = 82 - 48$

$17N = 34$

Divide both sides by 17

$17N/17 = 34/17$

$N = 34/17$

$N = 2$

Substitute 2 for N in $P = \frac{16 - 5N}{4}$

$P = \frac{16 - 5 * 2}{4}$

$P = \frac{16 - 10}{4}$

$P = \frac{6}{4}$

$P = 1.5$

Hence, a pen costs $1.50 and a notebook costs $2.00