Carina Begins To Solve The Equation $-4-\frac{2}{3} X=-6$ By Adding 4 To Both Sides. Which Statements Regarding The Rest Of The Solving Process Could Be True? Select Three Options.A. After Adding 4 To Both Sides, The Equation Is

Understanding the Problem

Carina is attempting to solve the equation $-4-\frac{2}{3} x=-6$ by adding 4 to both sides. This is a common strategy in algebra to isolate the variable and solve for its value. However, there are several ways to proceed with the solving process, and we will explore three possible options.

Option A: Adding 4 to Both Sides

After adding 4 to both sides, the equation becomes:

$$-\frac{2}{3} x = -2$$

This is a true statement, as adding 4 to both sides of the original equation results in this new equation.

Option B: Multiplying Both Sides by 3

To eliminate the fraction, Carina could multiply both sides of the equation by 3:

$$-2x = -6$$

This is also a true statement, as multiplying both sides of the equation by 3 results in this new equation.

Option C: Dividing Both Sides by -2

Alternatively, Carina could divide both sides of the equation by -2:

$$x = 3$$

This is also a true statement, as dividing both sides of the equation by -2 results in this new equation.

Conclusion

In conclusion, Carina's initial step of adding 4 to both sides of the equation is a valid strategy. However, the subsequent steps can vary depending on the approach chosen. The three options presented above demonstrate different ways to proceed with the solving process, and all of them are true statements.

Key Takeaways

• Adding 4 to both sides of the equation results in $-\frac{2}{3} x = -2$.
• Multiplying both sides of the equation by 3 results in $-2x = -6$.
• Dividing both sides of the equation by -2 results in $x = 3$.

By understanding these different approaches, students can develop a deeper understanding of linear equations and improve their problem-solving skills.

Additional Tips and Strategies

• When solving linear equations, it's essential to isolate the variable and solve for its value.
• Different approaches can be used to solve the same equation, and each approach has its own advantages and disadvantages.
• Practicing different strategies can help students develop a deeper understanding of linear equations and improve their problem-solving skills.

Common Mistakes to Avoid

• Failing to isolate the variable and solve for its value.
• Not considering different approaches to solving the equation.
• Not checking the validity of the solution.

Real-World Applications

• Linear equations are used in various real-world applications, such as physics, engineering, and economics.
• Understanding linear equations is essential for solving problems in these fields.
• Developing problem-solving skills in linear equations can help students succeed in these fields.

Conclusion

Frequently Asked Questions

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

A: The first step in solving a linear equation is to isolate the variable and solve for its value. This can be done by adding, subtracting, multiplying, or dividing both sides of the equation by a constant.

Q: How do I eliminate fractions in a linear equation?

A: To eliminate fractions in a linear equation, you can multiply both sides of the equation by the denominator of the fraction. For example, if the equation is $-\frac{2}{3} x = -2$, you can multiply both sides by 3 to get rid of the fraction.

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, on the other hand, 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 with a variable on both sides?

A: To solve a linear equation with a variable on both sides, you can use the distributive property to combine like terms. For example, if the equation is $2x + 3x = 5$, you can combine the like terms to get $5x = 5$.

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

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

1. Parentheses: Evaluate any expressions inside parentheses.
2. Exponents: Evaluate any exponential expressions.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Evaluate any addition and subtraction operations from left to right.

Q: How do I check the validity of a solution to a linear equation?

A: To check the validity of a solution to a linear equation, you can plug the solution back into the original equation and see if it is true. For example, if the solution to the equation $2x + 3 = 5$ is $x = 1$, you can plug $x = 1$ back into the original equation to get $2(1) + 3 = 5$, which is true.

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

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

• Failing to isolate the variable and solve for its value.
• Not considering different approaches to solving the equation.
• Not checking the validity of the solution.

Q: How do I apply linear equations to real-world problems?

A: Linear equations can be applied to a wide range of real-world problems, including physics, engineering, and economics. For example, a linear equation can be used to model the cost of producing a product, or to determine the amount of time it takes to complete a task.

Q: What are some tips for improving problem-solving skills in linear equations?

A: Some tips for improving problem-solving skills in linear equations include:

• Practicing different strategies for solving linear equations.
• Using visual aids, such as graphs and charts, to help understand the problem.
• Breaking down complex problems into simpler, more manageable parts.

Q: How do I know if a linear equation has a solution?

A: A linear equation has a solution if it is consistent, meaning that the equation is true for at least one value of the variable. If the equation is inconsistent, meaning that it is never true for any value of the variable, then it has no solution.

Q: What is the difference between a linear equation and a system of linear equations?

A: A linear equation is a single equation in which the highest power of the variable is 1. A system of linear equations, on the other hand, is a set of two or more linear equations that are solved simultaneously.