Nine Years Ago, Katie Was Twice As Old As Elena Was Then.Elena Realizes, In Four Years, I'll Be As Old As Katie Is Now!Elena Writes These Equations To Help Her Make Sense Of The Situation:$\begin{array}{l} k-9=2(e-9) \\ e+4=k \end{array}$If

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Introduction

In this article, we will delve into a mathematical puzzle presented by two individuals, Katie and Elena. Nine years ago, Katie was twice as old as Elena was then. Elena has realized that in four years, she will be as old as Katie is now. To make sense of this situation, Elena has written two equations to represent the relationship between their ages. In this article, we will solve these equations to determine the current ages of Katie and Elena.

The Equations

Elena has written the following two equations to represent the relationship between their ages:

k−9=2(e−9)e+4=k\begin{array}{l} k-9=2(e-9) \\ e+4=k \end{array}

where kk represents Katie's current age and ee represents Elena's current age.

Solving the First Equation

Let's start by solving the first equation:

k−9=2(e−9)k-9=2(e-9)

To solve for kk, we can add 9 to both sides of the equation:

k=2(e−9)+9k = 2(e-9) + 9

Expanding the right-hand side of the equation, we get:

k=2e−18+9k = 2e - 18 + 9

Simplifying the equation, we get:

k=2e−9k = 2e - 9

Solving the Second Equation

Now, let's solve the second equation:

e+4=ke+4=k

We can substitute the expression for kk from the first equation into this equation:

e+4=2e−9e+4=2e-9

Subtracting ee from both sides of the equation, we get:

4=e−94= e-9

Adding 9 to both sides of the equation, we get:

13=e13=e

Finding Katie's Age

Now that we have found Elena's age, we can substitute this value into the first equation to find Katie's age:

k=2e−9k = 2e - 9

Substituting e=13e=13 into this equation, we get:

k=2(13)−9k = 2(13) - 9

Expanding the right-hand side of the equation, we get:

k=26−9k = 26 - 9

Simplifying the equation, we get:

k=17k = 17

Conclusion

In this article, we have solved the equations written by Elena to determine the current ages of Katie and Elena. We have found that Elena is currently 13 years old and Katie is currently 17 years old. This solution has helped us to understand the relationship between their ages and has provided a clear answer to the puzzle presented by Katie and Elena.

Real-World Applications

This type of problem can be applied to real-world situations where we need to solve equations to determine unknown values. For example, in finance, we may need to solve equations to determine the value of an investment or the interest rate on a loan. In science, we may need to solve equations to determine the concentration of a substance or the rate of a chemical reaction.

Tips and Tricks

When solving equations, it's essential to follow the order of operations (PEMDAS) and to simplify the equation as much as possible. It's also crucial to check the solution by substituting the value back into the original equation.

Practice Problems

Here are some practice problems to help you apply the concepts learned in this article:

  1. Solve the equation x+3=2(x−2)x+3=2(x-2).
  2. Solve the equation y−4=3(y+1)y-4=3(y+1).
  3. Solve the equation z+2=2(z−3)z+2=2(z-3).

Conclusion

Introduction

In our previous article, we solved the equations written by Elena to determine the current ages of Katie and Elena. In this article, we will provide a Q&A section to help you understand the concepts and techniques used to solve equations.

Q: What is an equation?

A: An equation is a statement that two expressions are equal. It consists of two parts: the left-hand side (LHS) and the right-hand side (RHS). The LHS and RHS are separated by an equal sign (=).

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. PEMDAS stands for:

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next (e.g., 2^3).
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I simplify an equation?

A: To simplify an equation, we need to combine like terms and eliminate any unnecessary operations. Here are some steps to follow:

  1. Combine like terms: Combine any terms that have the same variable and coefficient.
  2. Eliminate unnecessary operations: Remove any operations that are not necessary to solve the equation.
  3. Simplify the equation: Simplify the equation by combining like terms and eliminating unnecessary operations.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation in which the highest power of the variable is 1. For example, 2x + 3 = 5 is a linear equation. A quadratic equation is an equation in which the highest power of the variable is 2. For example, x^2 + 4x + 4 = 0 is a quadratic equation.

Q: How do I solve a linear equation?

A: To solve a linear equation, we need to isolate the variable on one side of the equation. Here are some steps to follow:

  1. Add or subtract the same value to both sides of the equation to eliminate the constant term.
  2. Multiply or divide both sides of the equation by the same value to eliminate the coefficient of the variable.
  3. Simplify the equation and solve for the variable.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, we need to use the quadratic formula or factor the equation. Here are some steps to follow:

  1. Use the quadratic formula: The quadratic formula is x = (-b ± √(b^2 - 4ac)) / 2a, where a, b, and c are the coefficients of the quadratic equation.
  2. Factor the equation: Factor the quadratic equation into two binomials.

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

A: Here are some common mistakes to avoid when solving equations:

  1. Not following the order of operations (PEMDAS).
  2. Not simplifying the equation.
  3. Not isolating the variable on one side of the equation.
  4. Not checking the solution by substituting the value back into the original equation.

Conclusion

In this Q&A article, we have provided answers to some common questions about solving equations. We hope that this article has helped you understand the concepts and techniques used to solve equations and has inspired you to practice and apply these concepts in real-world situations.