Sharlene Sells Seashells By The Seashore. Blue Shells Cost $\$3$ And Pink Shells Cost $\$4$. Sharlene Sold 22 Shells And Made $\$76$. Which Of These Correctly Expresses The System Of Equations In Standard Form?a.

Introduction

Sharlene is a seashell seller who has been selling her wares by the seashore. She has two types of shells: blue shells that cost $\$3$ each and pink shells that cost $\$4$ each. On a particular day, Sharlene sold a total of 22 shells and made a total of $\$76$. We are asked to express this situation as a system of equations in standard form.

The System of Equations

Let's denote the number of blue shells sold as $b$ and the number of pink shells sold as $p$. We know that the total number of shells sold is 22, so we can write the first equation as:

$$b + p = 22$$

We also know that the total amount of money made from selling the shells is $\$76$. Since blue shells cost $\$3$ each and pink shells cost $\$4$ each, we can write the second equation as:

$$3b + 4p = 76$$

Expressing the System of Equations in Standard Form

A system of equations in standard form is written as:

$$\begin{cases} ax + by = c \\ dx + ey = f \end{cases}$$

where $a$, $b$, $c$, $d$, $e$, and $f$ are constants, and $x$ and $y$ are variables.

In our case, we have:

$$\begin{cases} b + p = 22 \\ 3b + 4p = 76 \end{cases}$$

To express this system of equations in standard form, we can rewrite the first equation as:

$$p = 22 - b$$

Substituting this expression for $p$ into the second equation, we get:

$$3b + 4(22 - b) = 76$$

Expanding and simplifying, we get:

$$3b + 88 - 4b = 76$$

Combine like terms:

$$-b + 88 = 76$$

Subtract 88 from both sides:

$$-b = -12$$

Multiply both sides by -1:

$$b = 12$$

Now that we have found the value of $b$, we can substitute it back into the first equation to find the value of $p$:

$$p = 22 - b$$

$$p = 22 - 12$$

$$p = 10$$

Conclusion

In this article, we have expressed the system of equations in standard form. We have found the values of $b$ and $p$ by solving the system of equations. The system of equations in standard form is:

$$\begin{cases} b + p = 22 \\ 3b + 4p = 76 \end{cases}$$

We have also found the values of $b$ and $p$ by solving the system of equations. The values of $b$ and $p$ are $b = 12$ and $p = 10$.

Frequently Asked Questions

• What is the system of equations in standard form?
• How do we express the system of equations in standard form?
• How do we solve the system of equations in standard form?

Step-by-Step Solution

1. Write the system of equations in standard form.
2. Solve the first equation for one variable.
3. Substitute the expression for the variable into the second equation.
4. Solve the resulting equation for the other variable.
5. Substitute the value of the variable back into the first equation to find the value of the other variable.

Final Answer

The final answer is $\boxed{\begin{cases} b + p = 22 \\ 3b + 4p = 76 \end{cases}}$.

Introduction

In our previous article, we explored the system of equations that describes Sharlene's seashell sales. We expressed the system of equations in standard form and solved for the values of $b$ and $p$. In this article, we will answer some frequently asked questions about the system of equations and provide a step-by-step solution to help you understand the concept better.

Q&A

Q: What is the system of equations in standard form?

A: The system of equations in standard form is a set of two or more equations that are written in the form:

$$\begin{cases} ax + by = c \\ dx + ey = f \end{cases}$$

where $a$, $b$, $c$, $d$, $e$, and $f$ are constants, and $x$ and $y$ are variables.

Q: How do we express the system of equations in standard form?

A: To express the system of equations in standard form, we need to rewrite the equations in the form:

$$\begin{cases} ax + by = c \\ dx + ey = f \end{cases}$$

We can do this by rearranging the terms and combining like terms.

Q: How do we solve the system of equations in standard form?

A: To solve the system of equations in standard form, we need to find the values of $x$ and $y$ that satisfy both equations. We can do this by using substitution or elimination methods.

Q: What is the difference between the substitution method and the elimination method?

A: The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. The elimination method involves adding or subtracting the equations to eliminate one variable.

Q: How do we choose which method to use?

A: We can choose which method to use based on the coefficients of the variables. If the coefficients are easy to work with, we can use the substitution method. If the coefficients are difficult to work with, we can use the elimination method.

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

A: Some common mistakes to avoid when solving systems of equations include:

• Not checking the solutions to make sure they satisfy both equations
• Not using the correct method to solve the system
• Not simplifying the equations before solving them
• Not checking for extraneous solutions

Step-by-Step Solution

1. Write the system of equations in standard form.
2. Choose a method to solve the system (substitution or elimination).
3. Solve one equation for one variable.
4. Substitute the expression for the variable into the other equation.
5. Solve the resulting equation for the other variable.
6. Check the solutions to make sure they satisfy both equations.
7. Simplify the equations before solving them.
8. Check for extraneous solutions.

Example

Let's use the system of equations from our previous article:

$$\begin{cases} b + p = 22 \\ 3b + 4p = 76 \end{cases}$$

We can solve this system using the substitution method.

1. Solve the first equation for $p$:

$$p = 22 - b$$

2. Substitute the expression for $p$ into the second equation:

$$3b + 4(22 - b) = 76$$

3. Simplify the equation:

$$3b + 88 - 4b = 76$$

4. Combine like terms:

$$-b + 88 = 76$$

5. Subtract 88 from both sides:

$$-b = -12$$

6. Multiply both sides by -1:

$$b = 12$$

7. Substitute the value of $b$ back into the first equation to find the value of $p$:

$$p = 22 - b$$

$$p = 22 - 12$$

$$p = 10$$

Conclusion

In this article, we have answered some frequently asked questions about the system of equations and provided a step-by-step solution to help you understand the concept better. We have also used the substitution method to solve the system of equations from our previous article.

Frequently Asked Questions

• What is the system of equations in standard form?
• How do we express the system of equations in standard form?
• How do we solve the system of equations in standard form?
• What is the difference between the substitution method and the elimination method?
• How do we choose which method to use?
• What are some common mistakes to avoid when solving systems of equations?

Final Answer

The final answer is $\boxed{\begin{cases} b + p = 22 \\ 3b + 4p = 76 \end{cases}}$.