The selling price of a desktop monitor, keyboard and mouse is $317.90 in total. During the Great Electronics Sale, two companies, $A$ and $B$, offered the following discounts to their customers.

Company	Discounts given for each item			Total price after the discount
	Desktop monitor	Keyboard	Mouse
$A$	10%	15%	10%	$282.66
$B$	5%	25%	20%	$283.72

1. Define variables and set up equations from the total price and Company A's discounted price.

Let the selling prices of the desktop monitor, keyboard and mouse be $d$, $k$ and $m$ respectively.

From the total price:
$d + k + m = 317.90$

Company A gives a 10% discount on the monitor, 15% on the keyboard, and 10% on the mouse. This means the customer pays 90%, 85%, and 90% of the original prices respectively:
$0.9d + 0.85k + 0.9m = 282.66$

2. Factor out the common coefficient in Company A's equation and use substitution to solve for the price of the keyboard ($k$).

Notice that $d$ and $m$ share the same coefficient ($0.9$) in Company A's equation:
$0.9(d + m) + 0.85k = 282.66$

From the first equation, express $(d + m)$ in terms of $k$:
$d + m = 317.90 - k$

Substitute this into the factored equation:
$0.9(317.90 - k) + 0.85k = 282.66$
$286.11 - 0.9k + 0.85k = 282.66$
$286.11 - 0.05k = 282.66$
$0.05k = 3.45$
$k = 69$

3. Form an equation using Company B's discounted price, substitute $k = 69$, and simplify it into a two-variable linear equation.

Company B gives a 5% discount on the monitor, 25% on the keyboard, and 20% on the mouse:
$0.95d + 0.75k + 0.8m = 283.72$

Substitute $k = 69$:
$0.95d + 0.75(69) + 0.8m = 283.72$
$0.95d + 51.75 + 0.8m = 283.72$
$0.95d + 0.8m = 231.97$

4. Solve the remaining simultaneous equations for $d$ and $m$.

Substitute $k = 69$ into the original total price equation:
$d + 69 + m = 317.90$
$d + m = 248.90 \implies m = 248.90 - d$

Substitute $m$ into the simplified Company B equation:
$0.95d + 0.8(248.90 - d) = 231.97$
$0.95d + 199.12 - 0.8d = 231.97$
$0.15d = 32.85$
$d = \frac{32.85}{0.15} = 219$

Calculate $m$:
$m = 248.90 - 219 = 29.90$

1. Find the original price of the mouse (from part a) by formulating a system of equations from the discount amounts.

Let the original prices of the monitor, keyboard, and mouse be $x$, $y$, and $z$ respectively.
The total discounts given by Company $A$ and Company $B$ are:
Company $A$: $317.90 - 282.66 = 35.24$
Company $B$: $317.90 - 283.72 = 34.18$

Formulating the discount equations:
$0.1x + 0.15y + 0.1z = 35.24$
$0.05x + 0.25y + 0.2z = 34.18$

From Company $A$'s discount:
$0.1(x + y + z) + 0.05y = 35.24$
Substitute $x + y + z = 317.90$:
$0.1(317.90) + 0.05y = 35.24$
$31.79 + 0.05y = 35.24 \implies 0.05y = 3.45 \implies y = 69$

Substitute $y = 69$ into Company $B$'s discount equation:
$0.05x + 0.25(69) + 0.2z = 34.18$
$0.05x + 17.25 + 0.2z = 34.18 \implies 0.05x + 0.2z = 16.93$
Multiply by $20$:
$x + 4z = 338.60$

We also know $x + z = 317.90 - y = 317.90 - 69 = 248.90$.
Subtracting this from the equation above:
$(x + 4z) - (x + z) = 338.60 - 248.90$
$3z = 89.70 \implies z = 29.90$
The original price of the mouse is $29.90.

2. Calculate the new total price at Company B with the additional employee discount.

The further discount is $2\%$ on the original price of the mouse:
$\text{Additional discount} = 0.02 \times 29.90 = \$0.598$

New total price Kim has to pay at Company $B$:
$283.72 - 0.598 = \$283.122 \approx \$283.12$

3. Compare the new price at Company B with Company A to conclude.

Since the new total price at Company $B$ ($283.12) is still greater than the total price at Company $A$ ($282.66), the additional discount does not make it more attractive to purchase from Company $B$.