Solve For { C $} . . . { \frac{6}{7}c + 9 = 51 \}

Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a specific linear equation, $\frac{6}{7}c + 9 = 51$, to demonstrate the step-by-step process involved. By the end of this article, you will have a clear understanding of how to solve linear equations and be able to apply this knowledge to various problems.

Understanding Linear Equations

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. Linear equations can be solved using various methods, including algebraic manipulation, graphing, and substitution.

The Given Equation

The given equation is $\frac{6}{7}c + 9 = 51$. This equation is a linear equation in one variable, $c$. Our goal is to solve for $c$.

Step 1: Isolate the Variable Term

To solve for $c$, we need to isolate the variable term, $\frac{6}{7}c$, on one side of the equation. We can do this by subtracting 9 from both sides of the equation.

# Given equation
equation = "6/7*c + 9 = 51"
new_equation = "6/7*c = 42"

Step 2: Eliminate the Fraction

The next step is to eliminate the fraction by multiplying both sides of the equation by the denominator, 7.

# Multiply both sides by 7
new_equation = "6*c = 294"

Step 3: Solve for c

Now that we have eliminated the fraction, we can solve for $c$ by dividing both sides of the equation by 6.

# Divide both sides by 6
c = 294 / 6

Step 4: Simplify the Expression

Finally, we can simplify the expression by evaluating the division.

# Simplify the expression
c = 49

Conclusion

In this article, we have demonstrated the step-by-step process of solving a linear equation, $\frac{6}{7}c + 9 = 51$. By following these steps, we were able to isolate the variable term, eliminate the fraction, and solve for $c$. This process can be applied to various linear equations, and with practice, you will become proficient in solving them.

Tips and Variations

• When solving linear equations, it is essential to follow the order of operations (PEMDAS) to ensure that the equation is solved correctly.
• In some cases, you may need to use inverse operations to isolate the variable term. For example, if the equation is $c + 9 = 51$, you would need to subtract 9 from both sides to isolate $c$.
• Linear equations can be solved using various methods, including graphing and substitution. However, algebraic manipulation is often the most efficient method.

Practice Problems

Try solving the following linear equations:

1. $2x + 5 = 11$
2. $\frac{3}{4}x - 2 = 7$
3. $x - 3 = 9$

Introduction

In our previous article, we demonstrated the step-by-step process of solving a linear equation, $\frac{6}{7}c + 9 = 51$. However, we understand that sometimes, it's not just about following a set of steps, but also about understanding the underlying concepts and being able to apply them to various problems. In this article, we will address some common questions and concerns related to solving linear equations.

Q: What is the difference between a linear equation and a quadratic 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. On the other hand, a quadratic equation is an equation in which the highest power of the variable(s) is 2. For example, $x^2 + 4x + 4 = 0$ is a quadratic equation.

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

A: To determine if an equation is linear or quadratic, you need to look at the highest power of the variable(s). If the highest power is 1, the equation is linear. If the highest power is 2, the equation is quadratic.

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that tells you which operations to perform first when solving an equation. The acronym PEMDAS stands for:

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

Q: How do I solve a linear equation with fractions?

A: To solve a linear equation with fractions, you need to eliminate the fractions by multiplying both sides of the equation by the denominator. For example, if the equation is $\frac{2}{3}x + 4 = 12$, you would multiply both sides by 3 to eliminate the fraction.

Q: What is the difference between a linear equation and a system of linear equations?

A: A linear equation is an equation in which the highest power of the variable(s) is 1. A system of linear equations is a set of two or more linear equations that are solved simultaneously. For example, $\begin{cases} 2x + 3y = 7 \\ x - 2y = -3 \end{cases}$ is a system of linear equations.

Q: How do I solve a system of linear equations?

A: To solve a system of linear equations, you can use various methods, including substitution, elimination, and graphing. The most common method is the substitution method, in which you solve one equation for one variable and then substitute that expression into the equation.

Conclusion

In this article, we have addressed some common questions and concerns related to solving linear equations. We hope that this Q&A guide has provided you with a better understanding of the concepts and methods involved in solving linear equations. Remember to practice regularly to become proficient in solving linear equations.

Practice Problems

Try solving the following linear equations:

1. $\frac{3}{4}x - 2 = 7$
2. $2x + 5 = 11$
3. $x - 3 = 9$

Remember to follow the steps outlined in this article to solve each equation.

Additional Resources

• Khan Academy: Linear Equations
• Mathway: Linear Equations
• Wolfram Alpha: Linear Equations

We hope that this Q&A guide has been helpful in your understanding of solving linear equations. If you have any further questions or concerns, please don't hesitate to ask.