Solving Mixture Problems Using Two Variables

Introduction

Mixture problems in algebra involve combining two or more solutions with different concentrations to achieve a desired concentration. These problems are commonly solved using systems of equations with two variables.

Key Concepts

• Assign Variables: Let x and y represent the unknown quantities (e.g., ounces of each solution).

• Create a Table: Organize the information about each solution, including volume, concentration, and amount of pure substance.

• Set Up Equations: Use the total volume and total amount of pure substance to write two equations.

• Solve the System: Use substitution or elimination to solve for the variables.

Example: Mixing Hydrochloric Acid Solutions

Problem: How many ounces of 5% hydrochloric acid and 20% hydrochloric acid must be combined to obtain 10 oz of solution that is 12.5% hydrochloric acid?

• Let x: ounces of 5% solution

• Let y: ounces of 20% solution

• Equation 1 (Total Volume):

• Equation 2 (Total Acid):

Solving the System

• Multiply Equation 2 by 100 to clear decimals:

• Use elimination:

  • Multiply Equation 1 by 5:

  • Subtract:

  • Substitute into Equation 1:

Solution

• 5 oz of 5% solution

• 5 oz of 20% solution

Verification

• Total acid:

• Concentration:

Solving Distance-Rate-Time Problems Using Two Variables

Introduction

Distance-rate-time problems involve finding unknown rates or times when two objects travel different distances at different speeds but share a common time or other relationship. These problems are solved using systems of equations.

Key Concepts

• Distance Formula:

• Assign Variables: Let x and y represent unknown rates or times.

• Set Up Equations: Use the relationships given in the problem to write two equations.

• Solve the System: Use substitution or elimination to find the unknowns.

Example: Car and Truck Travel Problem

Problem: A car travels 250 km in the same time that a truck travels 225 km. If the rate of the car is 8 km/hr faster than the rate of the truck, find both rates.

• Let x: rate of car (km/hr)

• Let y: rate of truck (km/hr)

• Relationship:

• Equal time:

Solving the System

• Substitute into the time equation:

  • Cross-multiply:

Solution

• Rate of car: 80 km/hr

• Rate of truck: 72 km/hr

Verification

• Time for car: hr

• Time for truck: hr

Both times are equal, confirming the solution.