Solve The Simultaneous Equations:$ \begin{align*} \log (x-2) + \log 2 &= 2 \log Y \\ \log (x-3y+3) &= 0 \end{align*} $

Introduction

Simultaneous 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 set of simultaneous equations involving logarithmic functions. We will break down the solution into manageable steps, making it easy to understand and follow.

Understanding the Equations

The given equations are:

$$\log (x-2) + \log 2 = 2 \log y$$

$$\log (x-3y+3) = 0$$

The first equation involves logarithmic functions with base 10, and the second equation is a simple logarithmic equation. Our goal is to solve for the variables x and y.

Step 1: Simplify the First Equation

To simplify the first equation, we can use the property of logarithms that states $\log a + \log b = \log (ab)$. Applying this property, we get:

$$\log (x-2) + \log 2 = \log ((x-2) \cdot 2)$$

$$\log (x-2) + \log 2 = \log (2x-4)$$

Now, we can rewrite the first equation as:

$$\log (2x-4) = 2 \log y$$

Step 2: Exponentiate Both Sides

To eliminate the logarithm, we can exponentiate both sides of the equation. Since the base of the logarithm is 10, we can use the property that $a^{\log_b x} = x$. Applying this property, we get:

$$(2x-4) = 10^{2 \log y}$$

$$2x-4 = 100y$$

Step 3: Simplify the Second Equation

The second equation is a simple logarithmic equation. We can rewrite it as:

$$x-3y+3 = 10^0$$

$$x-3y+3 = 1$$

Step 4: Solve for x and y

Now that we have simplified both equations, we can solve for x and y. We can start by solving the second equation for x:

$$x = 3y-2$$

Substituting this expression for x into the first equation, we get:

$$2(3y-2)-4 = 100y$$

$$6y-4-4 = 100y$$

$$6y-8 = 100y$$

$$-8 = 94y$$

$$y = -\frac{8}{94}$$

$$y = -\frac{4}{47}$$

Now that we have found the value of y, we can substitute it back into the expression for x:

$$x = 3\left(-\frac{4}{47}\right)-2$$

$$x = -\frac{12}{47}-2$$

$$x = -\frac{12}{47}-\frac{94}{47}$$

$$x = -\frac{106}{47}$$

Conclusion

In this article, we solved a set of simultaneous equations involving logarithmic functions. We broke down the solution into manageable steps, making it easy to understand and follow. By simplifying the equations and solving for x and y, we found the values of x and y to be $x = -\frac{106}{47}$ and $y = -\frac{4}{47}$, respectively.

Final Answer

The final answer is:

$x = -\frac{106}{47}$

$y = -\frac{4}{47}$

Additional Resources

For more information on solving simultaneous equations, check out the following resources:

• Khan Academy: Solving Simultaneous Equations
• Mathway: Solving Simultaneous Equations
• Wolfram Alpha: Solving Simultaneous Equations

FAQs

Q: What is the difference between simultaneous equations and linear equations? A: Simultaneous equations involve multiple equations with multiple variables, while linear equations involve a single equation with a single variable.

Q: How do I solve simultaneous equations with logarithmic functions? A: To solve simultaneous equations with logarithmic functions, you can use the properties of logarithms to simplify the equations and then solve for the variables.

Introduction

Simultaneous equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In our previous article, we solved a set of simultaneous equations involving logarithmic functions. In this article, we will provide a Q&A guide to help you understand and solve simultaneous equations.

Q: What are simultaneous equations?

A: Simultaneous equations are a set of equations that involve multiple variables and are solved together to find the values of the variables.

Q: What are the different types of simultaneous equations?

A: There are two main types of simultaneous equations:

• Linear simultaneous equations: These equations involve linear equations with multiple variables.
• Non-linear simultaneous equations: These equations involve non-linear equations with multiple variables.

Q: How do I solve simultaneous equations?

A: To solve simultaneous equations, you can use the following steps:

1. Simplify the equations: Use algebraic manipulations to simplify the equations.
2. Use substitution or elimination: Use substitution or elimination methods to solve for one variable.
3. Check the solution: Check the solution by plugging it back into the original equations.

Q: What are some common methods for solving simultaneous equations?

A: Some common methods for solving simultaneous equations include:

• Substitution method: Substitute one equation into another to solve for one variable.
• Elimination method: Add or subtract equations to eliminate one variable.
• Graphical method: Plot the equations on a graph to find the intersection point.

Q: How do I use the substitution method to solve simultaneous equations?

A: To use the substitution method, follow these steps:

1. Solve one equation for one variable: Solve one equation for one variable.
2. Substitute the expression into the other equation: Substitute the expression into the other equation.
3. Solve for the other variable: Solve for the other variable.

Q: How do I use the elimination method to solve simultaneous equations?

A: To use the elimination method, follow these steps:

1. Add or subtract equations: Add or subtract equations to eliminate one variable.
2. Solve for the remaining variable: Solve for the remaining variable.

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

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

• Not checking the solution: Not checking the solution by plugging it back into the original equations.
• Not using the correct method: Not using the correct method for the type of equation.
• Not simplifying the equations: Not simplifying the equations before solving.

Q: How do I check the solution to a simultaneous equation?

A: To check the solution to a simultaneous equation, follow these steps:

1. Plug the solution back into the original equations: Plug the solution back into the original equations.
2. Check if the equations are satisfied: Check if the equations are satisfied.

Conclusion

Solving simultaneous equations is a crucial skill for students and professionals alike. By understanding the different types of simultaneous equations, using the correct methods, and checking the solution, you can solve simultaneous equations with confidence. Remember to avoid common mistakes and to simplify the equations before solving.

Additional Resources

For more information on solving simultaneous equations, check out the following resources:

• Khan Academy: Solving Simultaneous Equations
• Mathway: Solving Simultaneous Equations
• Wolfram Alpha: Solving Simultaneous Equations

FAQs

Q: What is the difference between simultaneous equations and linear equations? A: Simultaneous equations involve multiple equations with multiple variables, while linear equations involve a single equation with a single variable.

Q: How do I solve simultaneous equations with logarithmic functions? A: To solve simultaneous equations with logarithmic functions, you can use the properties of logarithms to simplify the equations and then solve for the variables.

Q: What is the significance of solving simultaneous equations? A: Solving simultaneous equations is a crucial skill for students and professionals alike, as it is used in a wide range of applications, including physics, engineering, and economics.