Solve The Following Multi-step Equation: $2(z+5)+4=-12$ What Does $z$ Equal?

Introduction

In mathematics, multi-step equations are a type of algebraic equation that requires multiple steps to solve. These equations involve variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division. In this article, we will focus on solving a multi-step equation of the form $2(z+5)+4=-12$ and determine the value of the variable $z$.

Understanding the Equation

The given equation is $2(z+5)+4=-12$. To solve this equation, we need to follow the order of operations (PEMDAS) and simplify the expression step by step.

Step 1: Distribute the Coefficient

The first step is to distribute the coefficient 2 to the terms inside the parentheses.

2(z+5) = 2z + 10

So, the equation becomes:

2z + 10 + 4 = -12

Step 2: Combine Like Terms

Next, we combine the like terms on the left-hand side of the equation.

2z + 14 = -12

Step 3: Isolate the Variable

Now, we need to isolate the variable $z$ by subtracting 14 from both sides of the equation.

2z = -12 - 14

2z = -26

Step 4: Solve for $z$

Finally, we divide both sides of the equation by 2 to solve for $z$.

z = -26/2

z = -13

Conclusion

In this article, we solved a multi-step equation of the form $2(z+5)+4=-12$ and determined the value of the variable $z$. By following the order of operations and simplifying the expression step by step, we were able to isolate the variable and solve for its value. The final answer is $z = -13$.

Tips and Tricks

• When solving multi-step equations, it's essential to follow the order of operations (PEMDAS) and simplify the expression step by step.
• Use parentheses to group terms and make it easier to distribute coefficients.
• Combine like terms to simplify the equation and make it easier to solve.
• Isolate the variable by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.

Common Mistakes

• Failing to follow the order of operations (PEMDAS) and simplifying the expression step by step.
• Not using parentheses to group terms and distribute coefficients.
• Not combining like terms and simplifying the equation.
• Not isolating the variable by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.

Real-World Applications

Multi-step equations have numerous real-world applications in various fields such as physics, engineering, economics, and computer science. For example, in physics, multi-step equations are used to describe the motion of objects and calculate their velocities and accelerations. In engineering, multi-step equations are used to design and optimize systems and structures. In economics, multi-step equations are used to model and analyze economic systems and make predictions about future trends. In computer science, multi-step equations are used to develop algorithms and solve complex problems.

Practice Problems

1. Solve the equation $3(x-2)+5=-10$ and determine the value of the variable $x$.
2. Solve the equation $2(y+3)-4=-12$ and determine the value of the variable $y$.
3. Solve the equation $4(z-1)+2=-6$ and determine the value of the variable $z$.

Conclusion

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Introduction

In our previous article, we solved a multi-step equation of the form $2(z+5)+4=-12$ and determined the value of the variable $z$. In this article, we will provide a Q&A guide to help you understand and solve multi-step equations.

Q: What is a multi-step equation?

A: A multi-step equation is a type of algebraic equation that requires multiple steps to solve. These equations involve variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division.

Q: What are the steps to solve a multi-step equation?

A: The steps to solve a multi-step equation are:

1. Distribute the coefficient to the terms inside the parentheses.
2. Combine like terms to simplify the equation.
3. Isolate the variable by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.
4. Solve for the variable.

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 there are multiple operations in an expression. The acronym PEMDAS stands for:

• P: Parentheses
• E: Exponents
• M: Multiplication
• D: Division
• A: Addition
• S: Subtraction

Q: How do I distribute a coefficient to the terms inside the parentheses?

A: To distribute a coefficient to the terms inside the parentheses, you multiply the coefficient by each term inside the parentheses.

For example, if we have the equation $2(z+5)$, we would distribute the coefficient 2 to the terms inside the parentheses as follows:

$2(z+5) = 2z + 10$

Q: How do I combine like terms?

A: To combine like terms, you add or subtract the coefficients of the like terms.

For example, if we have the equation $2z + 3z$, we would combine the like terms as follows:

$2z + 3z = 5z$

Q: How do I isolate the variable?

A: To isolate the variable, you add, subtract, multiply, or divide both sides of the equation by the same value.

For example, if we have the equation $2z + 3 = 5$, we would isolate the variable by subtracting 3 from both sides of the equation:

$2z = 5 - 3$ $2z = 2$ $z = 1$

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

A: Some common mistakes to avoid when solving multi-step equations include:

• Failing to follow the order of operations (PEMDAS)
• Not using parentheses to group terms and distribute coefficients
• Not combining like terms and simplifying the equation
• Not isolating the variable by adding, subtracting, multiplying, or dividing both sides of the equation by the same value

Q: How do I check my answer?

A: To check your answer, you can plug the value of the variable back into the original equation and see if it is true.

For example, if we have the equation $2(z+5)+4=-12$ and we solve for $z$ and get $z = -13$, we can plug $z = -13$ back into the original equation to check our answer:

$2(-13+5)+4=-12$ $2(-8)+4=-12$ $-16+4=-12$ $-12=-12$

Since the equation is true, we know that our answer is correct.

Conclusion

In conclusion, solving multi-step equations requires a step-by-step approach and a thorough understanding of the order of operations (PEMDAS). By following these steps and simplifying the expression step by step, we can isolate the variable and solve for its value. Remember to avoid common mistakes and check your answer to ensure that it is correct.