How To Write A System Of Linear Equations

Let's dive into the heart of algebra and explore the art of constructing systems of linear equations. It's a foundational skill that unlocks a world of problem-solving potential, allowing you to model and analyze relationships between multiple variables.

Understanding Linear Equations

A linear equation, at its core, represents a straight line when graphed. Its general form is:

Ax + By = C

Where:

• x and y are variables, representing unknown quantities.
• A and B are coefficients, which are constant numbers multiplying the variables.
• C is a constant, also a number.

The key characteristic of a linear equation is that the variables are raised to the power of 1. There are no exponents, square roots, or other non-linear operations performed on them. This ensures the relationship between x and y remains a straight line.

Examples of Linear Equations:

• 2x + 3y = 7
• x - y = 5
• y = 4x - 1 (This can be rearranged to -4x + y = -1)

Non-Linear Equations (Examples):

• x² + y = 9 (x is squared)
• xy = 4 (x and y are multiplied)
• y = √x (x is under a square root)

What is a System of Linear Equations?

A system of linear equations is simply a collection of two or more linear equations involving the same variables. The goal is often to find the values of the variables that satisfy all the equations in the system simultaneously. This solution represents the point(s) where the lines represented by the equations intersect.

Example:

Consider the following system:

• 2x + y = 5
• x - y = 1

The solution to this system is x = 2 and y = 1, because these values satisfy both equations:

• 2(2) + 1 = 5
• 2 - 1 = 1

Why are Systems of Linear Equations Important?

They are used to model a wide variety of real-world situations, including:

• Mixture problems: Determining how much of different ingredients to combine to achieve a desired concentration.
• Investment problems: Calculating how much to invest in different accounts to reach a financial goal.
• Distance-rate-time problems: Solving for unknown speeds, distances, or times when objects are moving.
• Supply and demand: Analyzing the relationship between the price of a product and the quantity consumers are willing to buy.
• Network analysis: Modeling the flow of traffic in a transportation network or data in a computer network.

Steps to Write a System of Linear Equations from a Word Problem

The process of translating a word problem into a system of linear equations involves careful reading, identifying key information, and translating that information into mathematical expressions. Here's a step-by-step approach:

1. Read the Problem Carefully:

• Read the entire problem thoroughly. Don't just skim it. Understand the context, the question being asked, and the relationships between the different quantities.
• Identify what the problem is asking you to find. This will help you define your variables.

2. Define Your Variables:

• Assign variables to represent the unknown quantities you need to find. Use letters that are easy to remember and relate to the problem. For example, if the problem involves the number of apples and oranges, you could use a for the number of apples and o for the number of oranges.
• Clearly state what each variable represents. For example, "Let a = the number of apples" and "Let o = the number of oranges." This eliminates confusion later.

3. Identify Key Information and Relationships:

• Look for keywords and phrases that indicate mathematical operations:
  • "Sum," "total," "combined" indicate addition (+).
  • "Difference," "less than," "decreased by" indicate subtraction (-).
  • "Product," "times," "multiplied by" indicate multiplication (*).
  • "Quotient," "divided by," "ratio" indicate division (/).
  • "Is," "equals," "results in" indicate equality (=).
• Pay attention to units. Make sure you are comparing quantities with the same units (e.g., you can't add feet and inches without converting them to the same unit).
• Look for relationships between the variables. These relationships will form the basis of your equations.

4. Translate the Information into Equations:

• Use the information you identified in step 3 to write equations that relate the variables.
• Each equation should represent a specific piece of information from the problem.
• Make sure your equations are linear. The variables should only be raised to the power of 1.

5. Check Your Equations:

• Make sure your equations accurately reflect the information given in the problem.
• Substitute some possible values for the variables to see if the equations make sense.
• Ask yourself: "If the answer to this problem were [some value], would my equations still hold true?"

Example Word Problems and Their Corresponding Systems of Equations

Let's work through some examples to illustrate the process:

Example 1: The Ticket Sales Problem

Problem: A theater sold 800 tickets to a play. Adult tickets cost $8, and children's tickets cost $5. If the total revenue was $5200, how many of each type of ticket were sold?*

Solution:

1. Read the problem: We need to find the number of adult tickets and the number of children's tickets sold.

2. Define variables:

   • Let a = the number of adult tickets sold.
   • Let c = the number of children's tickets sold.

3. Identify key information and relationships:

   • Total tickets sold: 800
   • Price of adult ticket: $8
   • Price of children's ticket: $5
   • Total revenue: $5200

4. Translate the information into equations:

   • Equation 1 (Total tickets): a + c = 800
   • Equation 2 (Total revenue): 8a + 5c = 5200

System of Equations:

• a + c = 800
• 8a + 5c = 5200

Example 2: The Mixture Problem

Problem: A chemist needs to prepare 500 mL of a 25% acid solution. She has a 10% acid solution and a 40% acid solution in stock. How many milliliters of each solution should she mix to obtain the desired concentration?*

Solution:

1. Read the problem: We need to find the volume of the 10% solution and the volume of the 40% solution needed.

2. Define variables:

   • Let x = the volume (in mL) of the 10% acid solution.
   • Let y = the volume (in mL) of the 40% acid solution.

3. Identify key information and relationships:

   • Total volume: 500 mL
   • Concentration of solution 1: 10% (0.10)
   • Concentration of solution 2: 40% (0.40)
   • Desired concentration: 25% (0.25)

4. Translate the information into equations:

   • Equation 1 (Total volume): x + y = 500
   • Equation 2 (Acid content): 0.10x + 0.40y = 0.25(500) (which simplifies to 0.10x + 0.40y = 125)

System of Equations:

• x + y = 500
• 0.10x + 0.40y = 125

Example 3: The Investment Problem

Problem: Maria invests $10,000 in two different accounts. One account pays 5% annual interest, and the other pays 8% annual interest. If she earns a total of $680 in interest after one year, how much did she invest in each account?*

Solution:

1. Read the problem: We need to find the amount invested in each account.

2. Define variables:

   • Let x = the amount invested at 5% (in dollars).
   • Let y = the amount invested at 8% (in dollars).

3. Identify key information and relationships:

   • Total investment: $10,000
   • Interest rate on investment 1: 5% (0.05)
   • Interest rate on investment 2: 8% (0.08)
   • Total interest earned: $680

4. Translate the information into equations:

   • Equation 1 (Total investment): x + y = 10000
   • Equation 2 (Total interest): 0.05x + 0.08y = 680

System of Equations:

• x + y = 10000
• 0.05x + 0.08y = 680

Example 4: The Distance-Rate-Time Problem

Problem: Two cars leave the same city at the same time and travel in opposite directions. One car travels at 60 mph, and the other travels at 75 mph. How long will it take for them to be 540 miles apart?*

Solution:

1. Read the problem: We need to find the time it takes for the cars to be 540 miles apart.

2. Define variables:

   • Let t = the time (in hours) it takes for the cars to be 540 miles apart.
   • Let d1 = the distance traveled by the first car.
   • Let d2 = the distance traveled by the second car.

3. Identify key information and relationships:

   • Speed of car 1: 60 mph
   • Speed of car 2: 75 mph
   • Total distance apart: 540 miles
   • Distance = Rate * Time

4. Translate the information into equations:

   • Equation 1 (Distance of car 1): d1 = 60t
   • Equation 2 (Distance of car 2): d2 = 75t
   • Equation 3 (Total distance): d1 + d2 = 540

System of Equations:

• d1 = 60t
• d2 = 75t
• d1 + d2 = 540

(Note: In this case, we can simplify the system by substituting the first two equations into the third equation to get a single equation with one variable: 60t + 75t = 540)

General Tips for Writing Systems of Linear Equations:

• Read carefully and repeatedly: The most common mistake is misinterpreting the problem.
• Be organized: Clearly define your variables and write down all the given information.
• Look for hidden information: Sometimes, information is implied rather than explicitly stated.
• Practice, practice, practice: The more problems you solve, the better you will become at recognizing patterns and translating word problems into equations.
• Don't be afraid to ask for help: If you are stuck, ask a teacher, tutor, or classmate for assistance.

Solving Systems of Linear Equations

Once you have written your system of linear equations, you need to solve it to find the values of the variables. There are several methods for solving systems of linear equations, including:

• Graphing: Graphing each equation on the same coordinate plane and finding the point of intersection. This method is useful for visualizing the solution, but it is not always accurate, especially if the solution involves fractions or decimals.
• Substitution: Solving one equation for one variable and substituting that expression into the other equation. This method is useful when one of the equations can be easily solved for one variable.
• Elimination (also called Addition/Subtraction): Multiplying one or both equations by a constant so that the coefficients of one of the variables are opposites, then adding the equations together to eliminate that variable. This method is useful when the coefficients of one of the variables are already opposites or can be easily made opposites.
• Matrices: Using matrix operations to solve the system. This method is more advanced but can be very efficient for solving large systems of equations.

The choice of method depends on the specific system of equations and your personal preference. With practice, you will learn to recognize which method is most efficient for a given problem.

Conclusion

Writing systems of linear equations from word problems is a valuable skill that can be applied to a wide range of real-world situations. By following the steps outlined in this guide, you can learn to translate word problems into mathematical expressions and solve for the unknown quantities. Remember to read carefully, define your variables clearly, and practice regularly. With dedication and perseverance, you can master the art of writing and solving systems of linear equations.