What Is The First Step In Solving This Equation?$\[ 3(2x + 6) - 4x = 2(5x - 2) + 6 \\]A. Use The Subtraction Property Of Equality To Subtract 6 From Both Sides.B. Use The Addition Property Of Equality To Add \[$4x\$\] To Both Sides.C.

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Understanding the Equation

When solving an equation, it's essential to follow a step-by-step approach to ensure accuracy and clarity. The given equation is a linear equation that involves multiple operations, including multiplication, addition, and subtraction. To solve this equation, we need to follow the correct order of operations and apply the appropriate properties of equality.

The Equation

3(2x+6)−4x=2(5x−2)+6{ 3(2x + 6) - 4x = 2(5x - 2) + 6 }

Step 1: Simplify the Equation

The first step in solving this equation is to simplify it by applying the distributive property and combining like terms. This will help us to isolate the variable and make it easier to solve.

Distributive Property

The distributive property states that for any real numbers a, b, and c:

a(b + c) = ab + ac

We can apply this property to the given equation by distributing the numbers outside the parentheses to the terms inside.

Simplifying the Equation

Using the distributive property, we can simplify the equation as follows:

6x+18−4x=10x−4+6{ 6x + 18 - 4x = 10x - 4 + 6 }

Combining Like Terms

Now, we can combine like terms by adding or subtracting the coefficients of the same variables.

2x+18=10x+2{ 2x + 18 = 10x + 2 }

Isolating the Variable

The next step is to isolate the variable x by moving all the terms containing x to one side of the equation and the constant terms to the other side.

Subtracting 2x from Both Sides

To isolate the variable x, we need to subtract 2x from both sides of the equation.

2x−2x+18=10x−2x+2{ 2x - 2x + 18 = 10x - 2x + 2 }

This simplifies to:

18=8x+2{ 18 = 8x + 2 }

Subtracting 2 from Both Sides

Now, we need to subtract 2 from both sides of the equation to isolate the term containing x.

18−2=8x+2−2{ 18 - 2 = 8x + 2 - 2 }

This simplifies to:

16=8x{ 16 = 8x }

Dividing Both Sides by 8

Finally, we need to divide both sides of the equation by 8 to solve for x.

168=8x8{ \frac{16}{8} = \frac{8x}{8} }

This simplifies to:

2=x{ 2 = x }

Conclusion

In conclusion, the first step in solving this equation is to simplify it by applying the distributive property and combining like terms. This helps to isolate the variable and make it easier to solve. By following the correct order of operations and applying the appropriate properties of equality, we can solve the equation and find the value of x.

Answer

The correct answer is:

A. Use the subtraction property of equality to subtract 6 from both sides.

However, this is not the first step in solving the equation. The first step is to simplify the equation by applying the distributive property and combining like terms.

Discussion

What is the first step in solving this equation? Is it to simplify the equation by applying the distributive property and combining like terms, or is it to use the subtraction property of equality to subtract 6 from both sides? Share your thoughts and let's discuss this further.

Related Topics

  • Distributive Property
  • Combining Like Terms
  • Properties of Equality
  • Solving Linear Equations

References

Tags

  • Distributive Property
  • Combining Like Terms
  • Properties of Equality
  • Solving Linear Equations
  • Math
  • Algebra
  • Equations
    Q&A: Solving Linear Equations ==============================

Frequently Asked Questions

When solving linear equations, students often have questions about the correct steps to follow. Here are some frequently asked questions and their answers:

Q: What is the first step in solving a linear equation?

A: The first step in solving a linear equation is to simplify it by applying the distributive property and combining like terms. This helps to isolate the variable and make it easier to solve.

Q: How do I apply the distributive property?

A: To apply the distributive property, you need to multiply the number outside the parentheses to each term inside the parentheses. For example, in the equation 3(2x + 6), you would multiply 3 to each term inside the parentheses: 3(2x) + 3(6).

Q: What is the difference between the distributive property and combining like terms?

A: The distributive property is used to multiply a number outside the parentheses to each term inside the parentheses. Combining like terms is used to add or subtract the coefficients of the same variables. For example, in the equation 2x + 4x, you would combine the like terms by adding the coefficients: 6x.

Q: How do I isolate the variable in a linear equation?

A: To isolate the variable, you need to move all the terms containing the variable to one side of the equation and the constant terms to the other side. You can do this by adding or subtracting the same value to both sides of the equation.

Q: What is the order of operations when solving a linear equation?

A: The order of operations when solving a linear equation is:

  1. Simplify the equation by applying the distributive property and combining like terms.
  2. Isolate the variable by moving all the terms containing the variable to one side of the equation and the constant terms to the other side.
  3. Solve for the variable by dividing both sides of the equation by the coefficient of the variable.

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

A: Some common mistakes to avoid when solving linear equations include:

  • Not simplifying the equation by applying the distributive property and combining like terms.
  • Not isolating the variable by moving all the terms containing the variable to one side of the equation and the constant terms to the other side.
  • Not following the order of operations.

Q: How can I practice solving linear equations?

A: You can practice solving linear equations by working through examples and exercises in your textbook or online resources. You can also try solving linear equations on your own by creating your own problems and solutions.

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

A: Linear equations have many real-world applications, including:

  • Modeling population growth and decline
  • Calculating the cost of goods and services
  • Determining the amount of time it takes to complete a task
  • Solving problems in physics and engineering

Q: Can you provide some examples of linear equations?

A: Here are some examples of linear equations:

  • 2x + 3 = 5
  • x - 2 = 3
  • 4x + 2 = 10
  • x + 1 = 2

Q: How can I use technology to solve linear equations?

A: You can use technology, such as calculators or computer software, to solve linear equations. Many calculators and computer programs have built-in functions for solving linear equations, and can also graph the solutions.

Q: What are some common types of linear equations?

A: Some common types of linear equations include:

  • Simple linear equations: equations with one variable and a constant term.
  • Multi-variable linear equations: equations with multiple variables and constant terms.
  • Linear equations with fractions: equations with fractions as coefficients or constants.
  • Linear equations with decimals: equations with decimals as coefficients or constants.

Q: Can you provide some tips for solving linear equations?

A: Here are some tips for solving linear equations:

  • Read the equation carefully and identify the variable and constant terms.
  • Simplify the equation by applying the distributive property and combining like terms.
  • Isolate the variable by moving all the terms containing the variable to one side of the equation and the constant terms to the other side.
  • Follow the order of operations.
  • Check your solution by plugging it back into the original equation.

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

A: Some common mistakes to avoid when solving linear equations include:

  • Not simplifying the equation by applying the distributive property and combining like terms.
  • Not isolating the variable by moving all the terms containing the variable to one side of the equation and the constant terms to the other side.
  • Not following the order of operations.
  • Not checking your solution by plugging it back into the original equation.

Q: Can you provide some examples of linear equations with multiple variables?

A: Here are some examples of linear equations with multiple variables:

  • 2x + 3y = 5
  • x - 2y = 3
  • 4x + 2y = 10
  • x + 1y = 2

Q: How can I use linear equations to model real-world problems?

A: You can use linear equations to model real-world problems by identifying the variables and constant terms in the equation and using them to represent the problem. For example, if you are modeling the cost of goods and services, you can use a linear equation to represent the cost as a function of the number of goods or services.

Q: What are some common applications of linear equations in science and engineering?

A: Linear equations have many applications in science and engineering, including:

  • Modeling population growth and decline
  • Calculating the cost of goods and services
  • Determining the amount of time it takes to complete a task
  • Solving problems in physics and engineering

Q: Can you provide some examples of linear equations in science and engineering?

A: Here are some examples of linear equations in science and engineering:

  • Modeling the growth of a population: dP/dt = rP
  • Calculating the cost of goods and services: C = mx + b
  • Determining the amount of time it takes to complete a task: t = (d + v)/a
  • Solving problems in physics and engineering: F = ma, E = mc^2

Q: How can I use linear equations to solve problems in physics and engineering?

A: You can use linear equations to solve problems in physics and engineering by identifying the variables and constant terms in the equation and using them to represent the problem. For example, if you are solving a problem involving force and mass, you can use a linear equation to represent the force as a function of the mass.

Q: What are some common types of linear equations in physics and engineering?

A: Some common types of linear equations in physics and engineering include:

  • Simple linear equations: equations with one variable and a constant term.
  • Multi-variable linear equations: equations with multiple variables and constant terms.
  • Linear equations with fractions: equations with fractions as coefficients or constants.
  • Linear equations with decimals: equations with decimals as coefficients or constants.

Q: Can you provide some tips for solving linear equations in physics and engineering?

A: Here are some tips for solving linear equations in physics and engineering:

  • Read the equation carefully and identify the variables and constant terms.
  • Simplify the equation by applying the distributive property and combining like terms.
  • Isolate the variable by moving all the terms containing the variable to one side of the equation and the constant terms to the other side.
  • Follow the order of operations.
  • Check your solution by plugging it back into the original equation.

Q: What are some common mistakes to avoid when solving linear equations in physics and engineering?

A: Some common mistakes to avoid when solving linear equations in physics and engineering include:

  • Not simplifying the equation by applying the distributive property and combining like terms.
  • Not isolating the variable by moving all the terms containing the variable to one side of the equation and the constant terms to the other side.
  • Not following the order of operations.
  • Not checking your solution by plugging it back into the original equation.

Q: Can you provide some examples of linear equations in finance?

A: Here are some examples of linear equations in finance:

  • Modeling the growth of an investment: A = P(1 + r)^n
  • Calculating the cost of goods and services: C = mx + b
  • Determining the amount of time it takes to complete a task: t = (d + v)/a
  • Solving problems in finance: F = P(1 + r)^n

Q: How can I use linear equations to solve problems in finance?

A: You can use linear equations to solve problems in finance by identifying the variables and constant terms in the equation and using them to represent the problem. For example, if you are solving a problem involving the growth of an investment, you can use a linear equation to represent the growth as a function of the time period.

Q: What are some common types of linear equations in finance?

A: Some common types of linear equations in finance include:

  • Simple linear equations: equations with one variable and a constant term.
  • Multi-variable linear equations: equations with multiple variables and constant terms.
  • Linear equations with fractions: equations with fractions as coefficients or constants.
  • Linear equations with decimals: equations with decimals as coefficients or constants.

Q: Can you provide some tips for solving linear equations in finance?

A: Here are some tips for solving linear equations in finance:

  • Read the equation carefully and identify the variables and constant terms.
  • Simplify the equation by applying the distributive property and combining like terms.
  • Isolate the variable by moving all the terms containing the variable