Problem 35 - Algebra and Trigonometry

Question

Use the four-step strategy to solve each problem. Use \(x, y,\) and \(z\) to represent unknown quantities. Then translate from the verbal conditions of the problem to a system of three equations in three variables. On a recent trip to the convenience store, you picked up 2 gallons of milk, 5 bottles of water, and 6 snack-size bags of chips. Your total bill (before tax) was \(\$ 19.00 .\) If a bottle of water costs twice as much as a bag of chips, and a gallon of milk costs \(\$ 2.00\) more than a bottle of water, how much does each item cost?

Step-by-Step Solution

Answer

The price of a gallon of milk (x) is $3.50, a bottle of water (y) is $1.50, and a snack-size bag of chips (z) is $0.75.

Step 1: Formulate the Equations

The first equation is formed from the total bill: \(2x + 5y + 6z = 19\). Here, \(x\) represents the cost of milk, \(y\) the cost of water, and \(z\) the cost of a bag of chips. The second equation comes from the statement that a bottle of water costs twice as much as a bag of chips. This can be expressed as: \(y = 2z\). The third equation is derived from the information that a gallon of milk costs $2 more than a bottle of water, which gives the equation: \(x = y + 2\).

Step 2: Substitute Equations

Substitute the second equation into the first and third equations to eliminate the variable \(y\). This yields: \(2x + 10z + 6z = 19\) and \(x = 2z + 2\). Simplify these equations to get: \(2x + 16z = 19\) and \(x = 2z + 2\).

Step 3: Solve for z

Substitute the simplified third equation into the simplified first, which gives: \(2(2z + 2) + 16z = 19\). Simplify to find: \(4z + 4 + 16z = 19\) or \(20z + 4 = 19\). Solve for \(z\) to find \(z = 0.75\).

Step 4: Solve for y and x

Substitute the value of \(z\) into the second and third equations to find y and x. For \(y\), the equation is \(y = 2z\). With \(z = 0.75\), we find \(y = 2 * 0.75\) or \(y = 1.5\). For \(x\), the equation is \(x = y + 2\). With \(y = 1.5\), we find \(x = 1.5 + 2\) or \(x = 3.5\).

Key Concepts

Equation Formulation

When tackling problems that involve determining the cost of different items based on given conditions, it's essential to first translate the problem into mathematical equations. In this problem, we need to figure out the cost of three items: milk, water, and chips. To accomplish this, we assign variables:

• Let \( x \) be the cost of a gallon of milk.
• Let \( y \) be the cost of a bottle of water.
• Let \( z \) be the cost of a bag of chips.

Once the variables are defined, the next step is to create equations based on the information given:

• The total cost equation is based on the quantities and total bill: \( 2x + 5y + 6z = 19 \).
• Based on the relationship that water costs twice as much as chips: \( y = 2z \).
• With milk costing $2 more than water, we have: \( x = y + 2 \).

These equations represent the constraints of the problem and are used strategically to find the costs of each item.

Substitution Method

The substitution method is a reliable technique for solving systems of equations. It involves solving one of the equations for a single variable and then substituting that expression into the other equations. In this problem, we start by using the equation for the cost of water in terms of chips: \( y = 2z \).
We substitute \( y = 2z \) into the two other equations. The first substitution into the total cost equation gives us:

• \( 2x + 5(2z) + 6z = 19 \) simplifies to \( 2x + 10z + 6z = 19 \),
• which simplifies further to \( 2x + 16z = 19 \).

Next, substituting \( y = 2z \) into the equation for milk cost gives:

• \( x = 2z + 2 \).

Using substitution allows us to express the system with fewer variables, simplifying the calculation process and making it easier to solve for each unknown.

Cost Calculation

Calculating the cost of each item involves solving for one variable at a time and back-substituting to find the others. After substitution, we simplify our equations. For example, we take \( x = 2z + 2 \) and substitute into \( 2x + 16z = 19 \):

• The equation becomes \( 2(2z + 2) + 16z = 19 \).
• Upon simplifying: \( 4z + 4 + 16z = 19 \),
• which transforms to \( 20z + 4 = 19 \).

Solving this equation for \( z \) gives \( z = 0.75 \).
Once we have the value for \( z \), we can easily find the costs for water and milk:

• Using \( y = 2z \), substituting in \( z = 0.75 \) gives \( y = 1.5 \).
• For \( x = y + 2 \), substituting in \( y = 1.5 \) yields \( x = 3.5 \).

With these values, we determine each item's cost: \( \\(3.50 \) for milk, \( \\)1.50 \) for water, and \( \$0.75 \) for a bag of chips.

Three Variables Problem Solving

Solving problems with three variables can initially seem daunting, but it's manageable with a systematic approach. By following steps methodically, we ensure clarity and organize our solutions. The process is broken down into:

• Formulating equations to represent the problem mathematically.
• Employing substitution to reduce the number of variables in your equations.
• Solving for one variable and using back-substitution to find other variables.

Using our example, we started with three variables representing the costs. Each step ensured that our equations grew simpler:
By first reducing the problem to two variables, then focusing on solving the remaining linear equation, we efficiently solved for one unknown.
The final step involved substituting back into easier equations to find the remaining values.
Through this process, problems involving three variables become structured and systematic, making them easier to solve and understand.