Tamara Is Creating A Model Of A Rectangle. She Has 26 Inches Of Yellow Ribbon To Use For The Border Of The Rectangle. She Wants The Length, { L $}$, To Be 3 Inches Greater Than The Width, { W $}$.Which System Of Equations Could

Introduction

In mathematics, a system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables. In this article, we will explore a real-world scenario where Tamara is creating a model of a rectangle using 26 inches of yellow ribbon for the border. We will use a system of equations to find the length and width of the rectangle.

The Problem

Tamara has 26 inches of yellow ribbon to use for the border of the rectangle. She wants the length, { l $}$, to be 3 inches greater than the width, { w $}$. We can represent this relationship as an equation:

$$l = w + 3$$

The perimeter of the rectangle is equal to the sum of all its sides. Since the border is made of yellow ribbon, we can represent the perimeter as 26 inches. The formula for the perimeter of a rectangle is:

$$P = 2l + 2w$$

Substituting the value of $l$ from the first equation, we get:

$$26 = 2(w + 3) + 2w$$

Simplifying the Equation

To simplify the equation, we can expand the expression:

$$26 = 2w + 6 + 2w$$

Combine like terms:

$$26 = 4w + 6$$

Subtract 6 from both sides:

$$20 = 4w$$

Divide both sides by 4:

$$5 = w$$

Finding the Length

Now that we have found the value of $w$, we can substitute it into the first equation to find the value of $l$:

$$l = w + 3$$

$$l = 5 + 3$$

$$l = 8$$

The System of Equations

The system of equations that represents this problem is:

$$l = w + 3$$

$$26 = 2l + 2w$$

We can rewrite the second equation as:

$$26 = 2(w + 3) + 2w$$

This system of equations can be solved using substitution or elimination methods.

Conclusion

In this article, we have explored a real-world scenario where Tamara is creating a model of a rectangle using 26 inches of yellow ribbon for the border. We have used a system of equations to find the length and width of the rectangle. The system of equations is:

$$l = w + 3$$

$$26 = 2l + 2w$$

This system of equations can be solved using substitution or elimination methods.

Example Use Cases

This system of equations can be used in various real-world scenarios, such as:

• Designing a room with a specific shape and size
• Creating a model of a building or a structure
• Solving problems in physics and engineering

Tips and Tricks

When solving a system of equations, it's essential to:

• Read the problem carefully and understand what's being asked
• Identify the variables and the equations
• Choose the correct method to solve the system (substitution or elimination)
• Check the solution to ensure it satisfies both equations

By following these tips and tricks, you can become proficient in solving systems of equations and apply them to real-world problems.

Common Mistakes

When solving a system of equations, it's common to make mistakes such as:

• Not reading the problem carefully
• Not identifying the variables and the equations
• Choosing the wrong method to solve the system
• Not checking the solution

To avoid these mistakes, it's essential to:

• Read the problem carefully and understand what's being asked
• Identify the variables and the equations
• Choose the correct method to solve the system (substitution or elimination)
• Check the solution to ensure it satisfies both equations

Conclusion

Introduction

In our previous article, we explored a real-world scenario where Tamara is creating a model of a rectangle using 26 inches of yellow ribbon for the border. We used a system of equations to find the length and width of the rectangle. In this article, we will answer some frequently asked questions about systems of equations and provide additional examples and tips.

Q&A

Q: What is a system of equations?

A: A system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables.

Q: How do I know which method to use to solve a system of equations?

A: You can use either the substitution method or the elimination method to solve a system of equations. The substitution method involves substituting one equation into the other, while the elimination method involves adding or subtracting the equations to eliminate one of the variables.

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

A: The substitution method involves substituting one equation into the other, while the elimination method involves adding or subtracting the equations to eliminate one of the variables.

Q: How do I choose which variable to eliminate?

A: You can choose which variable to eliminate by looking at the coefficients of the variables in the equations. If the coefficients are the same, you can eliminate one of the variables by adding or subtracting the equations.

Q: What if I have a system of equations with three variables?

A: If you have a system of equations with three variables, you can use the elimination method to eliminate one of the variables, and then use the substitution method to solve for the remaining two variables.

Q: Can I use a graphing calculator to solve a system of equations?

A: Yes, you can use a graphing calculator to solve a system of equations. Graphing calculators can help you visualize the equations and find the point of intersection, which represents the solution to the system.

Q: How do I check my solution to a system of equations?

A: To check your solution to a system of equations, you can substitute the values of the variables into both equations and make sure that both equations are true.

Additional Examples

Example 1: Solving a System of Equations with Two Variables

Solve the system of equations:

$$x + y = 4$$

$$2x - y = 3$$

Solution

We can use the elimination method to solve this system of equations. First, we can multiply the first equation by 2 to get:

$$2x + 2y = 8$$

Then, we can add the two equations to eliminate the variable y:

$$4x = 11$$

Divide both sides by 4:

$$x = \frac{11}{4}$$

Now, we can substitute the value of x into one of the original equations to solve for y:

$$x + y = 4$$

$$\frac{11}{4} + y = 4$$

Subtract \frac{11}{4} from both sides:

$$y = 4 - \frac{11}{4}$$

Simplify:

$$y = \frac{7}{4}$$

Example 2: Solving a System of Equations with Three Variables

Solve the system of equations:

$$x + y + z = 6$$

$$2x - y + z = 3$$

$$x - 2y + z = 2$$

Solution

We can use the elimination method to solve this system of equations. First, we can multiply the first equation by 2 to get:

$$2x + 2y + 2z = 12$$

Then, we can add the second equation to the first equation to eliminate the variable x:

$$4x - y + 3z = 15$$

Now, we can multiply the third equation by 2 to get:

$$2x - 4y + 2z = 4$$

Then, we can add the third equation to the first equation to eliminate the variable x:

$$6y + 4z = 16$$

Divide both sides by 6:

$$y + \frac{2}{3}z = \frac{8}{3}$$

Now, we can substitute the value of y into one of the original equations to solve for z:

$$x + y + z = 6$$

$$x + \left(\frac{8}{3} - \frac{2}{3}z\right) + z = 6$$

Simplify:

$$x + \frac{8}{3} + \frac{1}{3}z = 6$$

Subtract \frac{8}{3} from both sides:

$$x + \frac{1}{3}z = \frac{10}{3}$$

Now, we can substitute the value of x into one of the original equations to solve for z:

$$2x - y + z = 3$$

$$2\left(\frac{10}{3} - \frac{1}{3}z\right) - \left(\frac{8}{3} - \frac{2}{3}z\right) + z = 3$$

Simplify:

$$\frac{20}{3} - \frac{2}{3}z - \frac{8}{3} + \frac{2}{3}z + z = 3$$

Combine like terms:

$$\frac{12}{3} + z = 3$$

Divide both sides by 3:

$$4 + z = 3$$

Subtract 4 from both sides:

$$z = -1$$

Now, we can substitute the value of z into one of the original equations to solve for y:

$$y + \frac{2}{3}z = \frac{8}{3}$$

$$y + \frac{2}{3}(-1) = \frac{8}{3}$$

Simplify:

$$y - \frac{2}{3} = \frac{8}{3}$$

Add \frac{2}{3} to both sides:

$$y = \frac{10}{3}$$

Now, we can substitute the value of y into one of the original equations to solve for x:

$$x + y + z = 6$$

$$x + \frac{10}{3} - 1 = 6$$

Simplify:

$$x + \frac{7}{3} = 6$$

Subtract \frac{7}{3} from both sides:

$$x = \frac{13}{3}$$

Conclusion

In this article, we have answered some frequently asked questions about systems of equations and provided additional examples and tips. We have also explored a real-world scenario where Tamara is creating a model of a rectangle using 26 inches of yellow ribbon for the border. By following the tips and tricks outlined in this article, you can become proficient in solving systems of equations and apply them to various scenarios. Remember to read the problem carefully, identify the variables and the equations, choose the correct method to solve the system, and check the solution to ensure it satisfies both equations.