Comparing the Original and Modified Equations

Oct 13, 2025 by miodeya.com 126 views

Let $p$ equal $3q + 4r^2 - 1$ . What happens to the value of $p$ when the value of $q$ is tripled and the value of $r$ is doubled?

In this problem, we are given an equation that defines the value of $p$ in terms of $q$ and $r$ . The equation is $p = 3q + 4r^2 - 1$ . We are asked to find out what happens to the value of $p$ when the value of $q$ is tripled and the value of $r$ is doubled.

Analyzing the Equation

To understand how the value of $p$ changes when the value of $q$ is tripled and the value of $r$ is doubled, we need to analyze the equation. The equation is a linear combination of $q$ and $r^2$ , with a constant term of $-1$ . When the value of $q$ is tripled, the term $3q$ becomes $9q$ . When the value of $r$ is doubled, the term $4r^2$ becomes $16r^2$ .

Applying the Changes

Now, let's apply the changes to the equation. When the value of $q$ is tripled and the value of $r$ is doubled, the equation becomes:

p = 9q + 16r^2 - 1

Comparing the Original and Modified Equations

Comparing the original equation $p = 3q + 4r^2 - 1$ with the modified equation $p = 9q + 16r^2 - 1$ , we can see that the coefficient of $q$ has increased by a factor of 3, and the coefficient of $r^2$ has increased by a factor of 4.

Determining the Effect on the Value of $p$

To determine the effect on the value of $p$ , we need to consider the changes in the coefficients of $q$ and $r^2$ . When the value of $q$ is tripled, the term $3q$ becomes $9q$ , which is 3 times larger than the original term. When the value of $r$ is doubled, the term $4r^2$ becomes $16r^2$ , which is 4 times larger than the original term.

Calculating the New Value of $p$

To calculate the new value of $p$ , we need to substitute the new values of $q$ and $r$ into the modified equation. Let's assume that the original value of $q$ is $q_0$ and the original value of $r$ is $r_0$ . Then, the new value of $q$ is $3q_0$ and the new value of $r$ is $2r_0$ .

Substituting these values into the modified equation, we get:

p = 9(3q_0) + 16(2r_0)^2 - 1

Simplifying the equation, we get:

p = 27q_0 + 64r_0^2 - 1

In conclusion, when the value of $q$ is tripled and the value of $r$ is doubled, the value of $p$ increases by a factor of 3 and 4, respectively. The new value of $p$ is given by the equation $p = 27q_0 + 64r_0^2 - 1$ , where $q_0$ and $r_0$ are the original values of $q$ and $r$ .

When the value of $q$ is tripled, the term $3q$ becomes $9q$ , which is 3 times larger than the original term.
When the value of $r$ is doubled, the term $4r^2$ becomes $16r^2$ , which is 4 times larger than the original term.
The new value of $p$ is given by the equation $p = 27q_0 + 64r_0^2 - 1$ , where $q_0$ and $r_0$ are the original values of $q$ and $r$ .

In this problem, we analyzed how the value of $p$ changes when the value of $q$ is tripled and the value of $r$ is doubled. We found that the value of $p$ increases by a factor of 3 and 4, respectively, and the new value of $p$ is given by the equation $p = 27q_0 + 64r_0^2 - 1$ . This problem demonstrates the importance of understanding how changes in variables affect the value of an expression.
Q&A: Let $p$ equal $3q + 4r^2 - 1$ . What happens to the value of $p$ when the value of $q$ is tripled and the value of $r$ is doubled?

Q: What is the original equation that defines the value of $p$ ? A: The original equation that defines the value of $p$ is $p = 3q + 4r^2 - 1$ .

Q: What happens to the value of $p$ when the value of $q$ is tripled? A: When the value of $q$ is tripled, the term $3q$ becomes $9q$ , which is 3 times larger than the original term.

Q: What happens to the value of $p$ when the value of $r$ is doubled? A: When the value of $r$ is doubled, the term $4r^2$ becomes $16r^2$ , which is 4 times larger than the original term.

Q: What is the new value of $p$ when the value of $q$ is tripled and the value of $r$ is doubled? A: The new value of $p$ is given by the equation $p = 27q_0 + 64r_0^2 - 1$ , where $q_0$ and $r_0$ are the original values of $q$ and $r$ .

Q: How does the value of $p$ change when the value of $q$ is tripled and the value of $r$ is doubled? A: The value of $p$ increases by a factor of 3 and 4, respectively, when the value of $q$ is tripled and the value of $r$ is doubled.

Q: What is the relationship between the original and modified equations? A: The modified equation $p = 9q + 16r^2 - 1$ is a multiple of the original equation $p = 3q + 4r^2 - 1$ , where the coefficients of $q$ and $r^2$ have been multiplied by 3 and 4, respectively.

Q: How can we use this problem to understand how changes in variables affect the value of an expression? A: This problem demonstrates the importance of understanding how changes in variables affect the value of an expression. By analyzing how the value of $p$ changes when the value of $q$ is tripled and the value of $r$ is doubled, we can gain a deeper understanding of how changes in variables can affect the value of an expression.

Q: What are some real-world applications of this problem? A: This problem has many real-world applications, such as understanding how changes in variables can affect the value of a function in physics, engineering, and economics.

Q: How can we use this problem to solve other problems? A: This problem can be used to solve other problems that involve understanding how changes in variables affect the value of an expression. By applying the same principles and techniques used in this problem, we can solve other problems that involve understanding how changes in variables can affect the value of an expression.

Q: What are some common mistakes to avoid when solving this problem? A: Some common mistakes to avoid when solving this problem include:

Not understanding how changes in variables affect the value of an expression
Not using the correct formulas and equations
Not checking the units and dimensions of the variables and expressions
Not using the correct mathematical operations and techniques

Q: How can we check our work and ensure that our solution is correct? A: We can check our work and ensure that our solution is correct by:

Using the correct formulas and equations
Checking the units and dimensions of the variables and expressions
Using the correct mathematical operations and techniques
Verifying that our solution satisfies the given conditions and constraints.