Question

Presale Order A wireless store owner takes presale orders for a new smartphone and tablet. He gets 340 preorders for the smartphone and 250 preorders for the tablet. The combined value of the preorders is \(\$ 486,000 .\) If the price of a smartphone and tablet together is \(\$ 1665,\) how much does each device cost?

Short answer

The price of a smartphone is \( 775 \) and the price of a tablet is \( 890 \).

Step-by-step solution

Define Variables

Let the price of a smartphone be denoted by \( x \) and the price of a tablet by \( y \. \) These are the unknowns we need to determine.

Set Up Equations

Using the information given, we can create two equations. The first equation comes from the combined value of the preorders: \[ 340x + 250y = 486000 \] The second equation comes from the price of a smartphone and a tablet together: \[ x + y = 1665 \]

Solve the Second Equation for One Variable

Solve the second equation for \( x \: \) \[ y = 1665 - x \]

Substitute into the First Equation

Substitute \( y = 1665 - x \) into the first equation: \ \[ 340x + 250(1665 - x) = 486000 \ \] Distribute and combine like terms: \ \[ 340x + 416250 - 250x = 486000 \ \]

Simplify the Equation

Combine like terms: \ \[ 90x + 416250 = 486000 \ \] Subtract 416250 from both sides: \ \[ 90x = 69750 \ \]

Solve for x

Divide both sides by 90: \ \[ x = 775 \ \] So the price of a smartphone is \( 775 \. \)

Solve for y

Substitute \ x = 775 \ back into the equation \ y = 1665 - x \: \ \[ y = 1665 - 775 \ \] \ \[ y = 890 \ \] So the price of a tablet is \( 890 \. \)

Key concepts

Solving Linear Equations

Solving linear equations involves finding the values of variables that make the equation true. In our problem, we have two equations that need to be solved simultaneously.
One equation is based on the total value of preorders, and the other is based on the combined price of a smartphone and a tablet.
This can feel confusing, but breaking it down step-by-step helps.

In our example:

• We defined the price of the smartphone as \( x \) and the price of the tablet as \( y \).
• The first equation comes from the total value of preorders: \[ 340x + 250y = 486000 \]
• The second equation comes from the combined price: \[ x + y = 1665 \]

Once we set up our equations, we use algebraic methods to solve them.
Simplify and solve each step methodically for clarity.

Substitution Method

The substitution method is a useful way to solve systems of linear equations, especially when dealing with two variables. It's like solving a puzzle by finding where the pieces fit.

First, solve one of the equations for one of the variables. In our problem, we solved the second equation for \( y \):

• \( y = 1665 - x \)

This means wherever we see \( y \) in our first equation, we can replace it with \( 1665 - x \).

Next, substitute this expression into the first equation:

• \[ 340x + 250 (1665 - x) = 486000 \]

Then simplify and solve the new equation. Combining terms and isolating variables allows for step-by-step simplification until we find \( x \).
Finally, we substitute \( x \) back into our expression for \( y \), giving the values of both variables.

This method is powerful for systematically working through systems of equations.

Real-World Application

Understanding how to solve systems of linear equations is not just an academic exercise; it has practical real-world applications.
For example, businesses need to determine the prices of products to maximize profit.

In our problem, a store owner needs to figure out the individual prices of a smartphone and a tablet based on preorders and total revenue.

• This means understanding the relationship between quantities and prices, which is critical for pricing strategy and financial planning.

Using mathematical models like linear equations helps organize and solve complex problems.

By learning these methods in the classroom, students can apply them to scenarios in business, science, engineering, and everyday life.
So, mastering these concepts has lasting value beyond just passing a test.