2 ( 5 - X ) = 19 -3 (x + 5 )

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Solving the Equation: 2(5 - x) = 19 - 3(x + 5)

In this article, we will delve into the world of algebra and solve a linear equation that involves parentheses and variables. The equation we will be working with is 2(5 - x) = 19 - 3(x + 5). This equation may seem daunting at first, but with the right steps and techniques, we can simplify it and find the value of the variable x.

Understanding the Equation

Before we start solving the equation, let's take a closer look at what it means. The equation is a linear equation, which means it is an equation that can be written in the form ax + b = c, where a, b, and c are constants. In this case, the equation is 2(5 - x) = 19 - 3(x + 5).

Step 1: Distribute the Numbers

To start solving the equation, we need to distribute the numbers inside the parentheses. This means we need to multiply the number outside the parentheses by each term inside the parentheses.

  • 2(5 - x) = 2(5) - 2(x)
  • 2(5 - x) = 10 - 2x

Step 2: Simplify the Right Side

Now that we have distributed the numbers, let's simplify the right side of the equation.

  • 19 - 3(x + 5) = 19 - 3x - 15
  • 19 - 3(x + 5) = 4 - 3x

Step 3: Set Up the Equation

Now that we have simplified both sides of the equation, let's set up the equation.

  • 10 - 2x = 4 - 3x

Step 4: Add 2x to Both Sides

To get rid of the variable x on the left side of the equation, we need to add 2x to both sides.

  • 10 = 4 - x

Step 5: Subtract 4 from Both Sides

To isolate the variable x, we need to subtract 4 from both sides.

  • 6 = -x

Step 6: Multiply Both Sides by -1

To make the variable x positive, we need to multiply both sides by -1.

  • -6 = x

And there you have it! We have solved the equation 2(5 - x) = 19 - 3(x + 5) and found the value of the variable x. The value of x is -6. This means that when we substitute x = -6 into the original equation, it will be true.

Here are some tips and tricks to help you solve equations like this one:

  • Always start by distributing the numbers inside the parentheses.
  • Simplify both sides of the equation before setting it up.
  • Use inverse operations to get rid of the variable x.
  • Check your work by substituting the value of x back into the original equation.

Here are some common mistakes to avoid when solving equations like this one:

  • Forgetting to distribute the numbers inside the parentheses.
  • Not simplifying both sides of the equation before setting it up.
  • Not using inverse operations to get rid of the variable x.
  • Not checking your work by substituting the value of x back into the original equation.

Solving equations like this one has many real-world applications. For example, in physics, we use equations to describe the motion of objects. In economics, we use equations to model the behavior of markets. In computer science, we use equations to optimize algorithms.

Q: What is the first step in solving the equation 2(5 - x) = 19 - 3(x + 5)?

A: The first step in solving the equation 2(5 - x) = 19 - 3(x + 5) is to distribute the numbers inside the parentheses. This means we need to multiply the number outside the parentheses by each term inside the parentheses.

Q: How do I distribute the numbers inside the parentheses?

A: To distribute the numbers inside the parentheses, we need to multiply the number outside the parentheses by each term inside the parentheses. For example, 2(5 - x) = 2(5) - 2(x) = 10 - 2x.

Q: What is the next step in solving the equation?

A: The next step in solving the equation is to simplify the right side of the equation. This means we need to combine like terms and simplify the expression.

Q: How do I simplify the right side of the equation?

A: To simplify the right side of the equation, we need to combine like terms and simplify the expression. For example, 19 - 3(x + 5) = 19 - 3x - 15 = 4 - 3x.

Q: What is the final step in solving the equation?

A: The final step in solving the equation is to isolate the variable x. This means we need to use inverse operations to get rid of the variable x.

Q: How do I isolate the variable x?

A: To isolate the variable x, we need to use inverse operations to get rid of the variable x. For example, if we have the equation 10 - 2x = 4 - 3x, we can add 2x to both sides to get 10 = 4 - x. Then, we can subtract 4 from both sides to get 6 = -x. Finally, we can multiply both sides by -1 to get x = -6.

Q: What is the value of the variable x?

A: The value of the variable x is -6.

Q: How do I check my work?

A: To check your work, you can substitute the value of x back into the original equation. If the equation is true, then you have solved it correctly.

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

A: Some common mistakes to avoid when solving equations like this one include:

  • Forgetting to distribute the numbers inside the parentheses.
  • Not simplifying both sides of the equation before setting it up.
  • Not using inverse operations to get rid of the variable x.
  • Not checking your work by substituting the value of x back into the original equation.

Q: What are some real-world applications of solving equations like this one?

A: Solving equations like this one has many real-world applications, including:

  • Physics: We use equations to describe the motion of objects.
  • Economics: We use equations to model the behavior of markets.
  • Computer Science: We use equations to optimize algorithms.

In conclusion, solving the equation 2(5 - x) = 19 - 3(x + 5) requires careful attention to detail and a solid understanding of algebraic techniques. By following the steps outlined in this article, you can solve equations like this one and apply the techniques to real-world problems.