Emmanuel And Mauricio Are Solving The Equation 7 ( X − 2 ) = 91 7(x-2)=91 7 ( X − 2 ) = 91 .Finish Solving Each Equation.[\begin{tabular}{|c|c|}\hline Emmanuel & Mauricio \\hline 7 ( X − 2 ) = 91 7(x-2)=91 7 ( X − 2 ) = 91 & 7 ( X − 2 ) = 91 7(x-2)=91 7 ( X − 2 ) = 91 \\hline 7 X − 14 = 91 7x-14=91 7 X − 14 = 91 & X − 2 = 13 X-2=13 X − 2 = 13

Mar 11, 2025 by ADMIN 345 views

**Solving Linear Equations: A Step-by-Step Guide**

Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will focus on solving linear equations, using the equation $7(x-2)=91$ as an example. We will guide you through the step-by-step process of solving this equation, highlighting the key concepts and techniques involved.

Step 1: Distributive Property

The first step in solving the equation $7(x-2)=91$ is to apply the distributive property. This property states that for any numbers $a$ , $b$ , and $c$ , the following equation holds:

$a(b+c) = ab + ac$

In this case, we can apply the distributive property to the left-hand side of the equation:

$7(x-2) = 7x - 14$

This step is crucial, as it allows us to simplify the equation and make it easier to solve.

Step 2: Isolate the Variable

Now that we have simplified the equation, we need to isolate the variable $x$ . To do this, we can add $14$ to both sides of the equation:

$7x - 14 + 14 = 91 + 14$

This simplifies to:

$7x = 105$

Step 3: Solve for x

Now that we have isolated the variable $x$ , we can solve for its value. To do this, we can divide both sides of the equation by $7$ :

$\frac{7x}{7} = \frac{105}{7}$

This simplifies to:

$x = 15$

Emmanuel's Solution

Emmanuel's solution to the equation $7(x-2)=91$ is as follows:

$7(x-2) = 91$

$7x - 14 = 91$

$7x = 105$

$x = 15$

Mauricio's Solution

Mauricio's solution to the equation $7(x-2)=91$ is as follows:

$7(x-2) = 91$

$x - 2 = 13$

$x = 15$

Discussion

As we can see, both Emmanuel and Mauricio arrived at the same solution to the equation $7(x-2)=91$ . This is because the steps involved in solving the equation are the same, regardless of the order in which they are performed.

However, it's worth noting that Mauricio's solution is slightly more concise, as he was able to isolate the variable $x$ in a single step. This is a great example of how different people can approach the same problem in different ways, but still arrive at the same solution.

Conclusion

Solving linear equations is a crucial skill for students to master, and it requires a combination of algebraic techniques and problem-solving strategies. In this article, we have guided you through the step-by-step process of solving the equation $7(x-2)=91$ , highlighting the key concepts and techniques involved.

Whether you are a student or a teacher, we hope that this article has provided you with a deeper understanding of linear equations and how to solve them. Remember, practice makes perfect, so be sure to try out some examples on your own to reinforce your understanding.

Additional Resources

If you are looking for additional resources to help you learn more about linear equations, we recommend checking out the following:

Khan Academy's Linear Equations course
Mathway's Linear Equations tutorial
IXL's Linear Equations practice exercises

These resources provide a wealth of information and practice exercises to help you master the art of solving linear equations.

Final Thoughts

Solving linear equations is a fundamental skill that is used in a wide range of mathematical and real-world applications. By mastering this skill, you will be able to tackle a wide range of problems and challenges, from simple algebraic equations to complex mathematical models.

We hope that this article has provided you with a deeper understanding of linear equations and how to solve them. Remember, practice makes perfect, so be sure to try out some examples on your own to reinforce your understanding.

Step 4: Check the Solution

Once we have solved for the value of $x$ , it's always a good idea to check our solution to make sure it's correct. We can do this by plugging the value of $x$ back into the original equation:

$7(x-2) = 91$

$7(15-2) = 91$

$7(13) = 91$

$91 = 91$

As we can see, the solution checks out, and we can be confident that our answer is correct.

Real-World Applications

Linear equations have a wide range of real-world applications, from physics and engineering to economics and finance. For example, linear equations can be used to model the motion of objects, the flow of fluids, and the growth of populations.

In physics, linear equations can be used to describe the motion of objects under the influence of gravity, friction, and other forces. For example, the equation $s = ut + \frac{1}{2}at^2$ describes the position of an object as a function of time, where $s$ is the position, $u$ is the initial velocity, $t$ is the time, and $a$ is the acceleration.

In engineering, linear equations can be used to design and optimize systems, such as bridges, buildings, and electrical circuits. For example, the equation $R = \frac{V}{I}$ describes the resistance of a circuit as a function of the voltage and current.

In economics, linear equations can be used to model the behavior of economic systems, such as supply and demand curves. For example, the equation $P = a + bQ$ describes the price of a good as a function of the quantity demanded, where $P$ is the price, $a$ is the intercept, and $b$ is the slope.

Conclusion

In conclusion, solving linear equations is a crucial skill that is used in a wide range of mathematical and real-world applications. By mastering this skill, you will be able to tackle a wide range of problems and challenges, from simple algebraic equations to complex mathematical models.

Final Thoughts

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable(s) is 1. In other words, it is an equation that can be written in the form:

ax + b = c

where a, b, and c are constants, and x is the variable.

Q: How do I solve a linear equation?

A: To solve a linear equation, you need to isolate the variable (x) on one side of the equation. You can do this by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.

Q: What is the distributive property?

A: The distributive property is a mathematical property that allows you to multiply a single value to multiple values inside parentheses. It is written as:

a(b + c) = ab + ac

Q: How do I use the distributive property to solve a linear equation?

A: To use the distributive property to solve a linear equation, you need to multiply the value outside the parentheses to each value inside the parentheses. For example:

2(x + 3) = 2x + 6

Q: What is the order of operations?

A: The order of operations is a set of rules that tells you which operations to perform first when solving an equation. The order of operations is:

Parentheses: Evaluate expressions inside parentheses first.
Exponents: Evaluate any exponential expressions next.
Multiplication and Division: Evaluate any multiplication and division operations from left to right.
Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I check my solution to a linear equation?

A: To check your solution to a linear equation, you need to plug the value of the variable back into the original equation and see if it is true. If it is true, then your solution is correct.

Q: What are some real-world applications of linear equations?

A: Linear equations have many real-world applications, including:

Physics: Linear equations can be used to describe the motion of objects under the influence of gravity, friction, and other forces.
Engineering: Linear equations can be used to design and optimize systems, such as bridges, buildings, and electrical circuits.
Economics: Linear equations can be used to model the behavior of economic systems, such as supply and demand curves.

Q: How can I practice solving linear equations?

A: There are many ways to practice solving linear equations, including:

Using online resources, such as Khan Academy or Mathway
Working with a tutor or teacher
Practicing with worksheets or exercises
Solving real-world problems that involve linear equations

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

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

Not following the order of operations
Not isolating the variable on one side of the equation
Not checking the solution to the equation
Not using the distributive property correctly

Q: How can I improve my skills in solving linear equations?

A: To improve your skills in solving linear equations, you can:

Practice regularly with worksheets or exercises
Work with a tutor or teacher to get individualized help
Use online resources, such as Khan Academy or Mathway
Solve real-world problems that involve linear equations

Conclusion

Additional Resources

If you are looking for additional resources to help you learn more about linear equations, we recommend checking out the following:

Khan Academy's Linear Equations course
Mathway's Linear Equations tutorial
IXL's Linear Equations practice exercises

These resources provide a wealth of information and practice exercises to help you master the art of solving linear equations.