Solve For { X $} . . . { \begin{array}{l} 5(6+x) = 90 \\ \end{array} \}

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 a specific type of linear equation, which is a simple equation with one variable. We will use the given equation as an example and walk through the steps to solve for the unknown variable.

The Given Equation

The given equation is:

$$5(6+x) = 90$$

This equation is a linear equation because it has only one variable, x, and it is in the form of ax = b, where a and b are constants.

Step 1: Distribute the Coefficient

To solve for x, we need to isolate the variable on one side of the equation. The first step is to distribute the coefficient 5 to the terms inside the parentheses.

$$5(6+x) = 5(6) + 5(x)$$

Using the distributive property, we can rewrite the equation as:

$$30 + 5x = 90$$

Step 2: Subtract 30 from Both Sides

The next step is to get rid of the constant term on the left side of the equation. We can do this by subtracting 30 from both sides of the equation.

$$30 + 5x - 30 = 90 - 30$$

Simplifying the equation, we get:

$$5x = 60$$

Step 3: Divide Both Sides by 5

Now that we have isolated the variable term, we can solve for x by dividing both sides of the equation by 5.

$$\frac{5x}{5} = \frac{60}{5}$$

Simplifying the equation, we get:

$$x = 12$$

Conclusion

In this article, we solved a simple linear equation using the distributive property and basic algebraic operations. We started with the given equation and walked through the steps to isolate the variable x. By following these steps, we were able to solve for x and find the value of the unknown variable.

Tips and Tricks

• When solving linear equations, always start by isolating the variable term on one side of the equation.
• Use the distributive property to simplify the equation and make it easier to solve.
• Be careful when subtracting or adding constants to both sides of the equation, as this can affect the value of the variable.

Real-World Applications

Linear equations have many real-world applications, including:

• Finance: Linear equations can be used to calculate interest rates, investment returns, and other financial metrics.
• Science: Linear equations can be used to model population growth, chemical reactions, and other scientific phenomena.
• Engineering: Linear equations can be used to design and optimize systems, such as bridges, buildings, and electronic circuits.

Common Mistakes

• Not isolating the variable term: Failing to isolate the variable term can make it difficult to solve the equation.
• Not using the distributive property: Failing to use the distributive property can make the equation more complicated and difficult to solve.
• Not checking the solution: Failing to check the solution can lead to incorrect answers.

Practice Problems

Try solving the following linear equations:

1. 2(x + 3) = 16
2. 4(x - 2) = 20
3. 3(x + 1) = 21

Answer Key

1. x = 6
2. x = 6
3. x = 6

Conclusion

Introduction

In our previous article, we discussed the basics of solving linear equations. In this article, we will provide a Q&A guide to help you better understand the concepts and apply them to real-world problems.

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable (x) is 1. It is a simple equation that can be solved using basic algebraic operations.

Q: What are the steps to solve a linear equation?

A: The steps to solve a linear equation are:

1. Distribute the coefficient to the terms inside the parentheses.
2. Isolate the variable term on one side of the equation.
3. Add or subtract constants to both sides of the equation.
4. Divide both sides of the equation by the coefficient.

Q: What is the distributive property?

A: The distributive property is a rule that allows us to multiply a coefficient to a term inside parentheses. It states that a(b + c) = ab + ac.

Q: How do I know if an equation is linear?

A: An equation is linear if it can be written in the form ax = b, where a and b are constants. If the equation has a variable raised to a power greater than 1, it is not linear.

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

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

• Not isolating the variable term
• Not using the distributive property
• Not checking the solution
• Adding or subtracting constants to the wrong side of the equation

Q: How do I check my solution?

A: To check your solution, plug the value of the variable back into the original equation and simplify. If the equation 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:

• Finance: Linear equations can be used to calculate interest rates, investment returns, and other financial metrics.
• Science: Linear equations can be used to model population growth, chemical reactions, and other scientific phenomena.
• Engineering: Linear equations can be used to design and optimize systems, such as bridges, buildings, and electronic circuits.

Q: Can I use linear equations to solve quadratic equations?

A: No, linear equations cannot be used to solve quadratic equations. Quadratic equations have a variable raised to a power of 2, which makes them more complex and requires different techniques to solve.

Q: What are some tips for solving linear equations?

A: Some tips for solving linear equations include:

• Start by isolating the variable term on one side of the equation.
• Use the distributive property to simplify the equation.
• Check your solution by plugging the value of the variable back into the original equation.
• Practice, practice, practice!

Q: Can I use a calculator to solve linear equations?

A: Yes, you can use a calculator to solve linear equations. However, it's always a good idea to check your solution by plugging the value of the variable back into the original equation.

Conclusion

Solving linear equations is a fundamental skill that is used in many areas of mathematics and science. By following the steps outlined in this article and practicing regularly, you can become proficient in solving linear equations and apply the concepts to real-world problems. Remember to always check your solution and use the distributive property to simplify the equation.