Select The Correct Location On The Table.Three Students Solved Linear Equations And Compared The Solutions. They Found That All Three Of The Equations Had The Same Solution.The Equations For Two Of The Students, Tristan And Isabelle, Are Shown Below.

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Introduction

In mathematics, solving linear equations is a fundamental concept that helps students develop problem-solving skills and understand the relationships between variables. When students are presented with a set of linear equations, they often try to find the solution that satisfies all the equations. In this article, we will explore the concept of solving linear equations and how students can compare their solutions to find the correct location on the table.

The Equations

Tristan and Isabelle, two students, were given the following linear equations to solve:

  • Tristan's equation: 2x + 5y = 11
  • Isabelle's equation: x + 3y = 7

Both students were asked to find the solution that satisfies both equations. After solving the equations, they found that both equations had the same solution.

Solving Linear Equations

To solve linear equations, students can use various methods such as substitution, elimination, or graphing. In this case, we will use the substitution method to solve the equations.

Substitution Method

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation.

Step 1: Solve one equation for one variable

Let's solve Tristan's equation for x:

2x + 5y = 11

Subtract 5y from both sides:

2x = 11 - 5y

Divide both sides by 2:

x = (11 - 5y) / 2

Step 2: Substitute the expression into the other equation

Now, substitute the expression for x into Isabelle's equation:

x + 3y = 7

Substitute x = (11 - 5y) / 2:

((11 - 5y) / 2) + 3y = 7

Step 3: Solve for y

Multiply both sides by 2 to eliminate the fraction:

11 - 5y + 6y = 14

Combine like terms:

y = 3

Step 4: Find the value of x

Now that we have the value of y, substitute it back into one of the original equations to find the value of x. Let's use Tristan's equation:

2x + 5y = 11

Substitute y = 3:

2x + 5(3) = 11

Simplify:

2x + 15 = 11

Subtract 15 from both sides:

2x = -4

Divide both sides by 2:

x = -2

The Solution

The solution to both equations is x = -2 and y = 3.

Comparing Solutions

Tristan and Isabelle compared their solutions and found that both equations had the same solution. This is because the equations are equivalent, meaning they represent the same relationship between the variables.

Equivalent Equations

Equivalent equations are equations that have the same solution. They can be obtained by multiplying or dividing both sides of an equation by a non-zero constant.

Example

Consider the following equations:

2x + 5y = 11

x + 3y = 7

These equations are equivalent because they can be obtained by multiplying or dividing both sides of one equation by a non-zero constant.

Conclusion

In this article, we explored the concept of solving linear equations and how students can compare their solutions to find the correct location on the table. We used the substitution method to solve the equations and found that both equations had the same solution. This is because the equations are equivalent, meaning they represent the same relationship between the variables. By understanding equivalent equations, students can develop problem-solving skills and improve their mathematical abilities.

Key Takeaways

  • Solving linear equations is a fundamental concept in mathematics.
  • Students can use various methods such as substitution, elimination, or graphing to solve linear equations.
  • Equivalent equations are equations that have the same solution.
  • Students can compare their solutions to find the correct location on the table.

Further Exploration

  • Explore other methods for solving linear equations, such as elimination or graphing.
  • Investigate the concept of equivalent equations and how they can be obtained.
  • Apply the concept of equivalent equations to real-world problems.

References

  • [1] "Linear Equations" by Math Open Reference
  • [2] "Equivalent Equations" by Khan Academy

Glossary

  • Linear Equation: An equation in which the highest power of the variable is 1.
  • Substitution Method: A method for solving linear equations by substituting one expression into another equation.
  • Equivalent Equations: Equations that have the same solution.
    Frequently Asked Questions: Selecting the Correct Location on the Table ====================================================================

Q: What is the main concept of solving linear equations?

A: The main concept of solving linear equations is to find the values of the variables that satisfy the equation. In this case, we are given two linear equations and we need to find the values of x and y that satisfy both equations.

Q: What is the substitution method for solving linear equations?

A: The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This method is useful when we have two equations with two variables.

Q: How do we know if two equations are equivalent?

A: Two equations are equivalent if they have the same solution. This means that if we solve one equation, we should get the same values for the variables as we get when we solve the other equation.

Q: What is the difference between equivalent equations and similar equations?

A: Equivalent equations are equations that have the same solution, while similar equations are equations that have the same form but may have different solutions.

Q: Can we use other methods to solve linear equations?

A: Yes, we can use other methods such as elimination or graphing to solve linear equations. However, the substitution method is a useful tool for solving equations with two variables.

Q: How do we apply the concept of equivalent equations to real-world problems?

A: We can apply the concept of equivalent equations to real-world problems by recognizing that equivalent equations represent the same relationship between variables. For example, in physics, we may have two equations that describe the motion of an object, and we need to find the values of the variables that satisfy both equations.

Q: What are some common mistakes to avoid when solving linear equations?

A: Some common mistakes to avoid when solving linear equations include:

  • Not checking if the equations are equivalent before solving them
  • Not using the correct method for solving the equations (e.g. substitution, elimination, or graphing)
  • Not checking if the solutions satisfy both equations

Q: How can we use technology to help us solve linear equations?

A: We can use technology such as calculators or computer software to help us solve linear equations. For example, we can use a graphing calculator to graph the equations and find the intersection points, or we can use computer software to solve the equations and find the values of the variables.

Q: What are some real-world applications of linear equations?

A: Linear equations have many real-world applications, including:

  • Physics: to describe the motion of objects
  • Engineering: to design and optimize systems
  • Economics: to model economic systems and make predictions
  • Computer Science: to solve problems and make decisions

Q: How can we extend our knowledge of linear equations to more complex equations?

A: We can extend our knowledge of linear equations to more complex equations by learning about quadratic equations, polynomial equations, and other types of equations. We can also learn about advanced techniques such as matrix algebra and differential equations.

Q: What are some resources for learning more about linear equations?

A: Some resources for learning more about linear equations include:

  • Textbooks and online resources such as Khan Academy and Math Open Reference
  • Online courses and tutorials such as Coursera and edX
  • Practice problems and worksheets to help us build our skills and confidence

Q: How can we apply our knowledge of linear equations to other areas of mathematics?

A: We can apply our knowledge of linear equations to other areas of mathematics such as algebra, geometry, and calculus. We can also use our knowledge of linear equations to solve problems in other fields such as physics, engineering, and economics.