Algebra

Mark and Brad bought the items shown below. Mark's Bill: 2 bicycles 3 scooters 5 iPad Total: $6450 Brad's Bill: 5 bicycles 6 scooters 8 iPad Total: $12150 Calculate the total cost of one bicycle, one doll and one iPad.

Step-by-step solution with explanation

Final Answer

The problem is underdetermined — with only 2 equations and 3 unknowns, there is no unique solution. A third independent equation (or one known price) is required to solve for b + s + i. As written, the answer cannot be determined.

Step-by-step solution

Set Up the System of Equations

We let b = bicycle price, s = scooter price, i = iPad price. We write one equation for Mark's bill and one for Brad's bill. The problem mentions 'doll' in the question but lists scooters on the bills, so the unknown item is a scooter (s).

Eliminate One Variable by Scaling

We multiply Mark's equation by 2 so that the scooter term (6s) matches Brad's equation. This lets us subtract to eliminate the scooters.

Subtract Brad's Equation from the Scaled Equation

Subtracting removes the 6s term. We're left with a simpler equation linking b and i.

Rewrite the Key Relationship

We isolate b in terms of i. This tells us the bicycle price depends on the iPad price. With only two equations and three unknowns, the system is underdetermined — there are infinitely many individual price combinations that satisfy both bills.

Find the Target Sum Directly

The question asks for the total cost of ONE bicycle + ONE scooter + ONE iPad. Let's see if we can find b + s + i directly from the two equations without needing each price separately.

Express Brad's Equation as a Multiple Plus Remainder

We substitute Mark's total into Brad's equation to find b − 2i. This confirms b − 2i = 12150 − 12900 = −750, matching Step 3.

Recognize the System Cannot Give a Unique Answer

With only 2 equations and 3 unknowns, there is no single numerical answer for b + s + i. The problem as stated is missing a third piece of information (a third bill or a known individual price) to find a unique solution. Note: the question also asks for a 'doll' which does not appear in either bill, making the problem inconsistent as written.

Understanding this problem

Learning Insight

A system of linear equations needs at least as many independent equations as unknowns to have a unique solution. With 3 unknowns (b, s, i) you need 3 equations. Two equations only let you express one variable in terms of another — you cannot pin down all three prices, or their sum, to a single value.

Quick Tip

Count your unknowns first! If you have 3 unknowns, look for 3 equations before trying to solve. If the counts don't match, flag the problem as underdetermined before doing any algebra.

Common Mistake

Students often assume they can find a unique answer even when there are fewer equations than unknowns. They pick one convenient value for a free variable and present it as the only solution — but that's just one of infinitely many valid solutions.