Solve The Equation Using The Principles Together. Don't Forget To Check Your Solution.$\[ 8(4x + 8) = 11 - (x + 2) \\]

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Introduction to Solving Equations

Solving equations is a fundamental concept in mathematics that involves finding the value of a variable that makes the equation true. In this article, we will guide you through the process of solving a complex equation using the principles of algebra. We will use the given equation as an example and walk you through each step of the solution process.

The Given Equation

The given equation is:

8(4x+8)=11−(x+2){ 8(4x + 8) = 11 - (x + 2) }

This equation involves parentheses, multiplication, and subtraction. To solve it, we need to follow the order of operations (PEMDAS) and simplify the equation step by step.

Step 1: Expand the Parentheses

The first step is to expand the parentheses on both sides of the equation. This involves multiplying the numbers outside the parentheses with the numbers inside.

8(4x+8)=32x+64{ 8(4x + 8) = 32x + 64 }

11−(x+2)=11−x−2=9−x{ 11 - (x + 2) = 11 - x - 2 = 9 - x }

So, the equation becomes:

32x+64=9−x{ 32x + 64 = 9 - x }

Step 2: Simplify the Equation

The next step is to simplify the equation by combining like terms. In this case, we can combine the x terms on the left side of the equation.

32x+x=33x{ 32x + x = 33x }

So, the equation becomes:

33x+64=9−x{ 33x + 64 = 9 - x }

Step 3: Isolate the Variable

The goal is to isolate the variable x on one side of the equation. To do this, we need to get rid of the constant term on the left side of the equation. We can do this by subtracting 64 from both sides of the equation.

33x=9−x−64{ 33x = 9 - x - 64 }

33x=−55−x{ 33x = -55 - x }

Step 4: Combine Like Terms

Now, we can combine the x terms on the left side of the equation.

33x+x=34x{ 33x + x = 34x }

So, the equation becomes:

34x=−55{ 34x = -55 }

Step 5: Solve for x

Finally, we can solve for x by dividing both sides of the equation by 34.

x=−5534{ x = \frac{-55}{34} }

x=−5534{ x = -\frac{55}{34} }

Conclusion

In this article, we solved a complex equation using the principles of algebra. We followed the order of operations (PEMDAS) and simplified the equation step by step. We expanded the parentheses, combined like terms, isolated the variable, and finally solved for x. The solution to the equation is x = -\frac{55}{34}.

Checking the Solution

To check our solution, we need to plug it back into the original equation and make sure it is true. Let's substitute x = -\frac{55}{34} into the original equation.

8(4(−5534)+8)=11−((−5534)+2){ 8(4(-\frac{55}{34}) + 8) = 11 - ((-\frac{55}{34}) + 2) }

8(−22034+8)=11−(−5534+2){ 8(-\frac{220}{34} + 8) = 11 - (-\frac{55}{34} + 2) }

8(−22034+27234)=11−(−5534+6834){ 8(-\frac{220}{34} + \frac{272}{34}) = 11 - (-\frac{55}{34} + \frac{68}{34}) }

8(5234)=11−(−1334){ 8(\frac{52}{34}) = 11 - (-\frac{13}{34}) }

41634=37434+1334{ \frac{416}{34} = \frac{374}{34} + \frac{13}{34} }

41634=38734{ \frac{416}{34} = \frac{387}{34} }

Since the left side of the equation is equal to the right side, our solution is correct.

Final Answer

The final answer to the equation is x = -\frac{55}{34}.

Introduction

Solving equations is a fundamental concept in mathematics that involves finding the value of a variable that makes the equation true. In this article, we will answer some frequently asked questions about solving equations, including tips and tricks for simplifying complex equations.

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 simplifying an expression. PEMDAS stands for Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction. When simplifying an expression, we need to follow the order of operations to ensure that we are performing the operations in the correct order.

Q: How do I simplify complex equations?

A: To simplify complex equations, we need to follow the order of operations (PEMDAS) and break down the equation into smaller parts. We can start by expanding the parentheses, combining like terms, and isolating the variable. We can also use algebraic properties, such as the distributive property and the commutative property, to simplify the equation.

Q: What is the distributive property?

A: The distributive property is a property of algebra that allows us to multiply a single term by multiple terms. For example, if we have the expression 2(x + 3), we can use the distributive property to expand it as 2x + 6.

Q: How do I isolate the variable?

A: To isolate the variable, we need to get rid of the constant term on the same side of the equation as the variable. We can do this by adding or subtracting the same value to both sides of the equation. For example, if we have the equation 2x + 3 = 5, we can isolate the variable by subtracting 3 from both sides of the equation.

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

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, we can use the quadratic formula, which is x = (-b ± √(b^2 - 4ac)) / 2a. We can also use factoring, which involves finding two numbers whose product is the constant term and whose sum is the coefficient of the variable.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that allows us to solve quadratic equations. It is x = (-b ± √(b^2 - 4ac)) / 2a, where a, b, and c are the coefficients of the quadratic equation.

Q: How do I check my solution?

A: To check our solution, we need to plug it back into the original equation and make sure it is true. We can also use algebraic properties, such as the distributive property and the commutative property, to check our solution.

Conclusion

Solving equations is a fundamental concept in mathematics that involves finding the value of a variable that makes the equation true. In this article, we answered some frequently asked questions about solving equations, including tips and tricks for simplifying complex equations. We also discussed the order of operations (PEMDAS), the distributive property, and the quadratic formula. By following these tips and tricks, we can simplify complex equations and solve for the value of the variable.

Final Tips

  • Always follow the order of operations (PEMDAS) when simplifying an expression.
  • Use algebraic properties, such as the distributive property and the commutative property, to simplify the equation.
  • Isolate the variable by getting rid of the constant term on the same side of the equation as the variable.
  • Use the quadratic formula to solve quadratic equations.
  • Check your solution by plugging it back into the original equation and making sure it is true.

By following these tips and tricks, we can simplify complex equations and solve for the value of the variable.