Choose A Linear Function For The Line Represented By The Point-slope Equation Y − 5 = 3 ( X − 2 Y - 5 = 3(x - 2 Y − 5 = 3 ( X − 2 ].A. F ( X ) = 3 X + 1 F(x) = 3x + 1 F ( X ) = 3 X + 1 B. F ( X ) = 3 X − 1 F(x) = 3x - 1 F ( X ) = 3 X − 1 C. F ( X ) = 8 X + 10 F(x) = 8x + 10 F ( X ) = 8 X + 10 D. F ( X ) = 8 X − 10 F(x) = 8x - 10 F ( X ) = 8 X − 10

Understanding the Point-Slope Equation

The point-slope equation is a mathematical representation of a line in the form of $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line and $m$ is the slope of the line. In the given equation $y - 5 = 3(x - 2)$, we can identify the point $(2, 5)$ and the slope $m = 3$. Our goal is to choose the correct linear function from the given options.

Analyzing the Options

Let's analyze each option to determine which one represents the line given by the point-slope equation.

Option A: $f(x) = 3x + 1$

To verify if this option is correct, we need to check if the point $(2, 5)$ satisfies the equation. Substituting $x = 2$ and $y = 5$ into the equation, we get:

$f(2) = 3(2) + 1 = 6 + 1 = 7$

However, the point $(2, 5)$ does not satisfy the equation $f(x) = 3x + 1$. Therefore, option A is not the correct linear function.

Option B: $f(x) = 3x - 1$

Let's check if the point $(2, 5)$ satisfies the equation. Substituting $x = 2$ and $y = 5$ into the equation, we get:

$f(2) = 3(2) - 1 = 6 - 1 = 5$

The point $(2, 5)$ satisfies the equation $f(x) = 3x - 1$. Therefore, option B is a potential candidate for the correct linear function.

Option C: $f(x) = 8x + 10$

To verify if this option is correct, we need to check if the point $(2, 5)$ satisfies the equation. Substituting $x = 2$ and $y = 5$ into the equation, we get:

$f(2) = 8(2) + 10 = 16 + 10 = 26$

However, the point $(2, 5)$ does not satisfy the equation $f(x) = 8x + 10$. Therefore, option C is not the correct linear function.

Option D: $f(x) = 8x - 10$

Let's check if the point $(2, 5)$ satisfies the equation. Substituting $x = 2$ and $y = 5$ into the equation, we get:

$f(2) = 8(2) - 10 = 16 - 10 = 6$

The point $(2, 5)$ does not satisfy the equation $f(x) = 8x - 10$. Therefore, option D is not the correct linear function.

Conclusion

Based on our analysis, only option B, $f(x) = 3x - 1$, satisfies the point-slope equation $y - 5 = 3(x - 2)$. Therefore, the correct linear function for the line represented by the point-slope equation is $f(x) = 3x - 1$.

Why is this important?

Choosing the correct linear function is crucial in mathematics, particularly in algebra and geometry. It helps us to accurately represent and analyze mathematical relationships, which is essential in various fields such as science, engineering, and economics. By understanding how to choose the correct linear function, we can develop a deeper appreciation for the underlying mathematical concepts and improve our problem-solving skills.

Real-World Applications

The concept of linear functions has numerous real-world applications. For example, in economics, linear functions can be used to model the relationship between the price of a product and its quantity demanded. In engineering, linear functions can be used to design and optimize systems, such as electrical circuits and mechanical systems. In computer science, linear functions can be used to develop algorithms for solving linear equations and inequalities.

Tips and Tricks

When working with linear functions, it's essential to remember the following tips and tricks:

• Always check if the point satisfies the equation.
• Use the slope-intercept form to rewrite the equation.
• Use the point-slope form to rewrite the equation.
• Use algebraic manipulations to simplify the equation.

By following these tips and tricks, you can develop a deeper understanding of linear functions and improve your problem-solving skills.

Conclusion

Frequently Asked Questions

Q: What is the point-slope equation?

A: The point-slope equation is a mathematical representation of a line in the form of $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line and $m$ is the slope of the line.

Q: How do I choose the correct linear function from the given options?

A: To choose the correct linear function, you need to check if the point satisfies the equation. You can do this by substituting the point into the equation and checking if the result is true.

Q: What if the point does not satisfy the equation?

A: If the point does not satisfy the equation, then the linear function is not the correct one. You need to try another option.

Q: Can I use the slope-intercept form to rewrite the equation?

A: Yes, you can use the slope-intercept form to rewrite the equation. This can help you to identify the correct linear function.

Q: Can I use the point-slope form to rewrite the equation?

A: Yes, you can use the point-slope form to rewrite the equation. This can help you to identify the correct linear function.

Q: What if I'm still unsure about which linear function is correct?

A: If you're still unsure, you can try using algebraic manipulations to simplify the equation. This can help you to identify the correct linear function.

Q: Why is choosing the correct linear function important?

A: Choosing the correct linear function is important because it helps you to accurately represent and analyze mathematical relationships. This is essential in various fields such as science, engineering, and economics.

Q: Can I use linear functions in real-world applications?

A: Yes, you can use linear functions in real-world applications. For example, in economics, linear functions can be used to model the relationship between the price of a product and its quantity demanded.

Q: What are some tips and tricks for working with linear functions?

A: Some tips and tricks for working with linear functions include:

• Always check if the point satisfies the equation.
• Use the slope-intercept form to rewrite the equation.
• Use the point-slope form to rewrite the equation.
• Use algebraic manipulations to simplify the equation.

Q: How can I improve my problem-solving skills when working with linear functions?

A: To improve your problem-solving skills when working with linear functions, you can practice solving problems and exercises. You can also try to identify the underlying mathematical concepts and relationships.

Common Mistakes to Avoid

Mistake 1: Not checking if the point satisfies the equation

A: Make sure to check if the point satisfies the equation before choosing the linear function.

Mistake 2: Not using the slope-intercept form to rewrite the equation

A: Try using the slope-intercept form to rewrite the equation to help you identify the correct linear function.

Mistake 3: Not using the point-slope form to rewrite the equation

A: Try using the point-slope form to rewrite the equation to help you identify the correct linear function.

Mistake 4: Not using algebraic manipulations to simplify the equation

A: Try using algebraic manipulations to simplify the equation to help you identify the correct linear function.

Conclusion

In conclusion, choosing the correct linear function is a critical skill in mathematics, particularly in algebra and geometry. By understanding how to choose the correct linear function, we can develop a deeper appreciation for the underlying mathematical concepts and improve our problem-solving skills. The concept of linear functions has numerous real-world applications, and by mastering this skill, we can unlock new opportunities in various fields.