Natalia Made A Mistake As She Solved The Following System Of Equations:${ \begin{array}{l} 5x - 2y = 15 \ y = 3x - 12 \end{array} }$What Did Natalia Do Well? What Should She Fix?

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Introduction

Solving systems of equations is a fundamental concept in mathematics, and it requires a clear understanding of algebraic manipulations and equation solving techniques. In this article, we will discuss a system of equations that Natalia attempted to solve, and we will identify what she did well and what she should fix.

The System of Equations

The system of equations that Natalia attempted to solve is given by:

{ \begin{array}{l} 5x - 2y = 15 \\ y = 3x - 12 \end{array} \}

What Natalia Did Well

Natalia did well in recognizing that the second equation is a linear equation in one variable, which can be solved for y in terms of x. This is a good start, as it allows her to substitute the expression for y into the first equation.

What Natalia Should Fix

However, Natalia should fix her approach by substituting the expression for y into the first equation correctly. Instead of substituting y = 3x - 12 into the first equation, she should substitute y = 3x - 12 into the first equation and then solve for x.

Step-by-Step Solution

To solve the system of equations, we will follow these steps:

  1. Substitute y = 3x - 12 into the first equation: We will substitute the expression for y into the first equation and then solve for x.

{ 5x - 2(3x - 12) = 15 \}

  1. Simplify the equation: We will simplify the equation by distributing the -2 to the terms inside the parentheses.

{ 5x - 6x + 24 = 15 \}

  1. Combine like terms: We will combine the like terms on the left-hand side of the equation.

{ -x + 24 = 15 \}

  1. Solve for x: We will solve for x by isolating the variable x on one side of the equation.

{ -x = 15 - 24 \}

{ -x = -9 \}

{ x = 9 \}

  1. Find the value of y: Now that we have found the value of x, we can substitute it into the second equation to find the value of y.

{ y = 3(9) - 12 \}

{ y = 27 - 12 \}

{ y = 15 \}

Conclusion

In conclusion, Natalia did well in recognizing that the second equation is a linear equation in one variable, which can be solved for y in terms of x. However, she should fix her approach by substituting the expression for y into the first equation correctly and then solving for x. By following the step-by-step solution outlined above, we can solve the system of equations and find the values of x and y.

Common Mistakes to Avoid

When solving systems of equations, there are several common mistakes to avoid:

  • Not substituting the expression for y into the first equation correctly: This can lead to incorrect solutions or no solution at all.
  • Not simplifying the equation: This can lead to incorrect solutions or no solution at all.
  • Not combining like terms: This can lead to incorrect solutions or no solution at all.
  • Not solving for x correctly: This can lead to incorrect solutions or no solution at all.

Tips and Tricks

When solving systems of equations, here are some tips and tricks to keep in mind:

  • Read the problem carefully: Make sure you understand what the problem is asking for.
  • Identify the type of equation: Determine whether the equation is linear, quadratic, or another type of equation.
  • Use substitution or elimination: Choose the method that is most suitable for the problem.
  • Simplify the equation: Combine like terms and simplify the equation as much as possible.
  • Solve for x correctly: Isolate the variable x on one side of the equation.

Real-World Applications

Solving systems of equations has many real-world applications, including:

  • Physics and engineering: Solving systems of equations is used to model real-world problems, such as the motion of objects or the behavior of electrical circuits.
  • Economics: Solving systems of equations is used to model economic systems, such as supply and demand or the behavior of markets.
  • Computer science: Solving systems of equations is used in computer science to solve problems, such as linear programming or graph theory.

Conclusion

Introduction

Solving systems of equations is a fundamental concept in mathematics that requires a clear understanding of algebraic manipulations and equation solving techniques. In this article, we will provide a Q&A guide to help you understand and solve systems of equations.

Q: What is a system of equations?

A: A system of equations is a set of two or more equations that are related to each other through a common variable or variables.

Q: How do I solve a system of equations?

A: To solve a system of equations, you can use either the substitution method or the elimination method. The substitution method involves substituting one equation into the other, while the elimination method involves adding or subtracting the equations to eliminate one of the variables.

Q: What is the substitution method?

A: The substitution method involves substituting one equation into the other. For example, if you have two equations:

{ x + y = 5 \}

{ x - y = 3 \}

You can substitute the second equation into the first equation to get:

{ (x - y) + y = 5 \}

{ x = 5 \}

Q: What is the elimination method?

A: The elimination method involves adding or subtracting the equations to eliminate one of the variables. For example, if you have two equations:

{ x + y = 5 \}

{ x - y = 3 \}

You can add the two equations to eliminate the variable y:

{ (x + y) + (x - y) = 5 + 3 \}

{ 2x = 8 \}

Q: How do I choose between the substitution method and the elimination method?

A: You can choose between the substitution method and the elimination method based on the type of equations you are working with. If the equations are linear and have the same variable, the elimination method may be easier to use. If the equations are non-linear or have different variables, the substitution method may be more suitable.

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

A: Some common mistakes to avoid when solving systems of equations include:

  • Not reading the problem carefully
  • Not identifying the type of equation
  • Not using the correct method (substitution or elimination)
  • Not simplifying the equation
  • Not solving for the correct variable

Q: How do I check my answer when solving a system of equations?

A: To check your answer when solving a system of equations, you can substitute the values of the variables back into the original equations to make sure they are true.

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

A: Solving systems of equations has many real-world applications, including:

  • Physics and engineering: Solving systems of equations is used to model real-world problems, such as the motion of objects or the behavior of electrical circuits.
  • Economics: Solving systems of equations is used to model economic systems, such as supply and demand or the behavior of markets.
  • Computer science: Solving systems of equations is used in computer science to solve problems, such as linear programming or graph theory.

Q: How can I practice solving systems of equations?

A: You can practice solving systems of equations by working through examples and exercises in a textbook or online resource. You can also try solving systems of equations on your own using real-world problems or scenarios.

Conclusion

In conclusion, solving systems of equations is a fundamental concept in mathematics that requires a clear understanding of algebraic manipulations and equation solving techniques. By following the Q&A guide outlined above, you can learn how to solve systems of equations and apply this knowledge to real-world problems.