Use The Steps For Solving Linear Equations To Solve The Given Linear Equation For $x$.$\frac{12x - 13}{14} + \frac{x + 10}{28} = \frac{17}{14}$

$$

Introduction to Linear Equations

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. A linear equation is an equation in which the highest power of the variable(s) is 1. In this article, we will focus on solving a linear equation that involves fractions. The given equation is $\frac{12x - 13}{14} + \frac{x + 10}{28} = \frac{17}{14}$. Our goal is to solve for the variable $x$.

Step 1: Multiply Both Sides of the Equation by the Least Common Multiple (LCM) of the Denominators

To eliminate the fractions, we need to multiply both sides of the equation by the least common multiple (LCM) of the denominators. In this case, the denominators are 14, 28, and 14. The LCM of these numbers is 28. Multiplying both sides of the equation by 28, we get:

$$28 \times \left(\frac{12x - 13}{14} + \frac{x + 10}{28}\right) = 28 \times \frac{17}{14}$$

Step 2: Distribute the Multiplication to Each Term

Now, we need to distribute the multiplication to each term inside the parentheses. This will eliminate the fractions and make it easier to solve for $x$.

$$28 \times \frac{12x - 13}{14} + 28 \times \frac{x + 10}{28} = 28 \times \frac{17}{14}$$

Step 3: Simplify Each Term

Simplifying each term, we get:

$$2(12x - 13) + (x + 10) = 2 \times 17$$

Step 4: Distribute the Multiplication to Each Term

Now, we need to distribute the multiplication to each term inside the parentheses.

$$24x - 26 + x + 10 = 34$$

Step 5: Combine Like Terms

Combining like terms, we get:

$$25x - 16 = 34$$

Step 6: Add 16 to Both Sides of the Equation

To isolate the term with $x$, we need to add 16 to both sides of the equation.

$$25x = 34 + 16$$

Step 7: Simplify the Right-Hand Side of the Equation

Simplifying the right-hand side of the equation, we get:

$$25x = 50$$

Step 8: Divide Both Sides of the Equation by 25

Finally, we need to divide both sides of the equation by 25 to solve for $x$.

$$x = \frac{50}{25}$$

Conclusion

In this article, we have walked through the steps to solve a linear equation that involves fractions. By following these steps, we have successfully solved for the variable $x$. The given equation was $\frac{12x - 13}{14} + \frac{x + 10}{28} = \frac{17}{14}$. Our goal was to solve for $x$, and we have achieved that by multiplying both sides of the equation by the LCM of the denominators, distributing the multiplication to each term, simplifying each term, combining like terms, adding 16 to both sides of the equation, simplifying the right-hand side of the equation, and finally dividing both sides of the equation by 25. The solution to the equation is $x = 2$.

Final Answer

The final answer is $\boxed{2}$.

$$

Introduction to Linear Equations

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. A linear equation is an equation in which the highest power of the variable(s) is 1. In this article, we will focus on solving a linear equation that involves fractions. The given equation is $\frac{12x - 13}{14} + \frac{x + 10}{28} = \frac{17}{14}$. Our goal is to solve for the variable $x$.

Q&A: Solving Linear Equations

Q: What is the first step in solving a linear equation that involves fractions?

A: The first step in solving a linear equation that involves fractions is to multiply both sides of the equation by the least common multiple (LCM) of the denominators. This will eliminate the fractions and make it easier to solve for $x$.

Q: How do I find the LCM of the denominators?

A: To find the LCM of the denominators, you need to list the multiples of each denominator and find the smallest multiple that is common to all of them. In this case, the denominators are 14, 28, and 14. The LCM of these numbers is 28.

Q: What is the next step after multiplying both sides of the equation by the LCM?

A: After multiplying both sides of the equation by the LCM, you need to distribute the multiplication to each term inside the parentheses. This will eliminate the fractions and make it easier to solve for $x$.

Q: How do I simplify each term after distributing the multiplication?

A: To simplify each term, you need to multiply each term inside the parentheses by the LCM. For example, if you have the term $\frac{12x - 13}{14}$, you would multiply it by 28 to get $2(12x - 13)$.

Q: What is the next step after simplifying each term?

A: After simplifying each term, you need to combine like terms. This will help you to isolate the term with $x$.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of the terms that have the same variable. For example, if you have the terms $24x$ and $x$, you would combine them to get $25x$.

Q: What is the final step in solving a linear equation?

A: The final step in solving a linear equation is to divide both sides of the equation by the coefficient of the term with $x$. This will give you the value of $x$.

Q: What if I have a fraction on the right-hand side of the equation?

A: If you have a fraction on the right-hand side of the equation, you need to multiply both sides of the equation by the denominator of the fraction. This will eliminate the fraction and make it easier to solve for $x$.

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

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

Conclusion

In this article, we have walked through the steps to solve a linear equation that involves fractions. We have also answered some common questions that students may have when solving linear equations. By following these steps and using the tips and tricks provided, you should be able to solve linear equations with ease.

Final Answer

The final answer is $\boxed{2}$.