Solutions to Practice Problems

Oct 13, 2025 by recoveryshake.com 68 views

Understanding the Problem

To solve for d, we need to isolate the variable d on one side of the equation. The given equation is d ÷ 2 = -108. Our goal is to find the value of d.

Isolating the Variable

To isolate the variable d, we need to get rid of the division by 2 on the left side of the equation. We can do this by multiplying both sides of the equation by 2.

Multiplying Both Sides

When we multiply both sides of the equation by 2, we get:

d ÷ 2 × 2 = -108 × 2

This simplifies to:

d = -216

Verifying the Solution

To verify our solution, we can plug the value of d back into the original equation and check if it's true.

Substituting the Value of d

Substituting d = -216 into the original equation, we get:

-216 ÷ 2 = -108

This is indeed true, so our solution is correct.

Conclusion

In this problem, we were given the equation d ÷ 2 = -108 and asked to solve for d. We isolated the variable d by multiplying both sides of the equation by 2, which gave us d = -216. We then verified our solution by plugging the value of d back into the original equation.

Tips and Tricks

When solving equations, it's essential to isolate the variable on one side of the equation.
Multiplying both sides of an equation by a non-zero number is a common technique for isolating variables.
Always verify your solution by plugging the value of the variable back into the original equation.

Real-World Applications

Solving equations is a fundamental skill that has numerous real-world applications. For example, in physics, equations are used to describe the motion of objects and predict their behavior. In finance, equations are used to calculate interest rates and investment returns. In engineering, equations are used to design and optimize systems.

Common Mistakes

Failing to isolate the variable on one side of the equation.
Multiplying both sides of an equation by a zero number, which would result in an undefined value.
Not verifying the solution by plugging the value of the variable back into the original equation.

Practice Problems

Solve for x in the equation x ÷ 3 = 12.
Solve for y in the equation y ÷ 4 = -20.
Solve for z in the equation z ÷ 2 = 15.

Solutions to Practice Problems

x ÷ 3 = 12 x = 12 × 3 x = 36
y ÷ 4 = -20 y = -20 × 4 y = -80
z ÷ 2 = 15 z = 15 × 2 z = 30

Conclusion

Solving equations is a fundamental skill that has numerous real-world applications. By isolating the variable on one side of the equation and verifying the solution, we can find the value of the variable. With practice, we can become proficient in solving equations and apply this skill to a wide range of problems.

Frequently Asked Questions

Q: What is the first step in solving the equation d ÷ 2 = -108?

A: The first step in solving the equation d ÷ 2 = -108 is to isolate the variable d on one side of the equation. We can do this by multiplying both sides of the equation by 2.

Q: Why do we multiply both sides of the equation by 2?

A: We multiply both sides of the equation by 2 to get rid of the division by 2 on the left side of the equation. This allows us to isolate the variable d.

Q: What is the value of d in the equation d ÷ 2 = -108?

A: The value of d in the equation d ÷ 2 = -108 is -216.

Q: How do we verify the solution?

A: We verify the solution by plugging the value of d back into the original equation and checking if it's true.

Q: What is the importance of verifying the solution?

A: Verifying the solution is essential to ensure that our answer is correct. If we don't verify the solution, we may end up with an incorrect answer.

Q: Can we use other methods to solve the equation d ÷ 2 = -108?

A: Yes, we can use other methods to solve the equation d ÷ 2 = -108. For example, we can multiply both sides of the equation by 2/2, which is equivalent to multiplying by 1.

Q: What is the result of multiplying both sides of the equation d ÷ 2 = -108 by 2/2?

A: The result of multiplying both sides of the equation d ÷ 2 = -108 by 2/2 is d = -216.

Q: Can we use a calculator to solve the equation d ÷ 2 = -108?

A: Yes, we can use a calculator to solve the equation d ÷ 2 = -108. However, it's essential to understand the underlying math and not just rely on the calculator.

Q: What is the difference between solving an equation and solving an inequality?

A: Solving an equation involves finding the value of the variable that makes the equation true, while solving an inequality involves finding the range of values that make the inequality true.

Q: Can we solve an inequality using the same methods as solving an equation?

A: No, we cannot solve an inequality using the same methods as solving an equation. Inequalities require different techniques and strategies.

Q: What is the importance of understanding the concept of variables and constants?

A: Understanding the concept of variables and constants is essential in solving equations and inequalities. Variables are the unknown values we're trying to find, while constants are the known values.

Q: Can we use variables and constants in real-world applications?

A: Yes, we can use variables and constants in real-world applications. For example, in physics, we use variables and constants to describe the motion of objects and predict their behavior.

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, while a quadratic equation is an equation in which the highest power of the variable is 2.

Q: Can we solve a quadratic equation using the same methods as solving a linear equation?

A: No, we cannot solve a quadratic equation using the same methods as solving a linear equation. Quadratic equations require different techniques and strategies.

Q: What is the importance of understanding the concept of functions?

A: Understanding the concept of functions is essential in solving equations and inequalities. Functions describe the relationship between variables and constants.

Q: Can we use functions in real-world applications?

A: Yes, we can use functions in real-world applications. For example, in economics, we use functions to describe the relationship between variables and constants.

Q: What is the difference between a function and a relation?

A: A function is a relation in which each input corresponds to exactly one output, while a relation is a set of ordered pairs.

Q: Can we use relations in real-world applications?

A: Yes, we can use relations in real-world applications. For example, in computer science, we use relations to describe the relationship between variables and constants.

Q: What is the importance of understanding the concept of domain and range?

A: Understanding the concept of domain and range is essential in solving equations and inequalities. Domain refers to the set of possible input values, while range refers to the set of possible output values.

Q: Can we use domain and range in real-world applications?

A: Yes, we can use domain and range in real-world applications. For example, in engineering, we use domain and range to describe the relationship between variables and constants.

Q: What is the difference between a discrete function and a continuous function?

A: A discrete function is a function in which the input values are distinct and separate, while a continuous function is a function in which the input values are continuous and can take on any value.

Q: Can we use discrete and continuous functions in real-world applications?

A: Yes, we can use discrete and continuous functions in real-world applications. For example, in physics, we use discrete and continuous functions to describe the motion of objects and predict their behavior.

Q: What is the importance of understanding the concept of limits?

A: Understanding the concept of limits is essential in solving equations and inequalities. Limits describe the behavior of a function as the input values approach a certain value.

Q: Can we use limits in real-world applications?

A: Yes, we can use limits in real-world applications. For example, in engineering, we use limits to describe the behavior of a system as the input values approach a certain value.

Q: What is the difference between a limit and a derivative?

A: A limit is a value that a function approaches as the input values approach a certain value, while a derivative is a measure of how fast a function changes as the input values change.

Q: Can we use limits and derivatives in real-world applications?

A: Yes, we can use limits and derivatives in real-world applications. For example, in physics, we use limits and derivatives to describe the motion of objects and predict their behavior.

Q: What is the importance of understanding the concept of integrals?

A: Understanding the concept of integrals is essential in solving equations and inequalities. Integrals describe the accumulation of a quantity over a given interval.

Q: Can we use integrals in real-world applications?

A: Yes, we can use integrals in real-world applications. For example, in engineering, we use integrals to describe the accumulation of a quantity over a given interval.

Q: What is the difference between a definite integral and an indefinite integral?

A: A definite integral is an integral that has a specific upper and lower bound, while an indefinite integral is an integral that does not have a specific upper and lower bound.

Q: Can we use definite and indefinite integrals in real-world applications?

A: Yes, we can use definite and indefinite integrals in real-world applications. For example, in physics, we use definite and indefinite integrals to describe the motion of objects and predict their behavior.

Q: What is the importance of understanding the concept of differential equations?

A: Understanding the concept of differential equations is essential in solving equations and inequalities. Differential equations describe the rate of change of a quantity over time.

Q: Can we use differential equations in real-world applications?

A: Yes, we can use differential equations in real-world applications. For example, in physics, we use differential equations to describe the motion of objects and predict their behavior.

Q: What is the difference between a linear differential equation and a nonlinear differential equation?

A: A linear differential equation is a differential equation in which the highest power of the derivative is 1, while a nonlinear differential equation is a differential equation in which the highest power of the derivative is greater than 1.

Q: Can we use linear and nonlinear differential equations in real-world applications?

A: Yes, we can use linear and nonlinear differential equations in real-world applications. For example, in physics, we use linear and nonlinear differential equations to describe the motion of objects and predict their behavior.

Q: What is the importance of understanding the concept of eigenvalues and eigenvectors?

A: Understanding the concept of eigenvalues and eigenvectors is essential in solving equations and inequalities. Eigenvalues and eigenvectors describe the behavior of a system under a linear transformation.

Q: Can we use eigenvalues and eigenvectors in real-world applications?

A: Yes, we can use eigenvalues and eigenvectors in real-world applications. For example, in engineering, we use eigenvalues and eigenvectors to describe the behavior of a system under a linear transformation.

Q: What is the difference between a symmetric matrix and a skew-symmetric matrix?

A: A symmetric matrix is a matrix that is equal to its transpose, while a skew-symmetric matrix is a matrix that is equal to the negative of its transpose.

Q: Can we use symmetric and skew-symmetric matrices in real-world applications?

A: Yes, we can use symmetric and skew-symmetric matrices in real-world applications. For example, in physics, we use symmetric and skew-symmetric matrices to describe the behavior of a system under a linear transformation.

Q: What is the importance of understanding the concept of singular value decomposition?

A: Understanding the concept of singular value decomposition is essential in solving equations and inequalities. Singular value decomposition describes the behavior of a matrix under a linear transformation.

Q: Can we use singular value decomposition in real-world applications?

A: Yes, we can use singular value decomposition in real-world applications. For example, in engineering, we use singular value decomposition