What Is The Equation Of The Line That Is Parallel To The Line Y − 1 = 4 ( X + 3 Y - 1 = 4(x + 3 Y − 1 = 4 ( X + 3 ] And Passes Through The Point ( 4 , 32 (4, 32 ( 4 , 32 ]?A. Y = − 1 4 X + 33 Y = -\frac{1}{4} X + 33 Y = − 4 1 ​ X + 33 B. Y = − 1 4 X + 36 Y = -\frac{1}{4} X + 36 Y = − 4 1 ​ X + 36 C. Y = 4 X − 16 Y = 4x - 16 Y = 4 X − 16 D. $y

Introduction

In mathematics, particularly in the realm of geometry and algebra, the concept of parallel lines plays a crucial role. Two lines are said to be parallel if they lie in the same plane and never intersect, no matter how far they are extended. In this article, we will delve into the world of parallel lines and explore the equation of a line that is parallel to a given line and passes through a specific point.

Understanding the Given Line

The given line is represented by the equation $y - 1 = 4(x + 3)$. To simplify this equation, we can expand the right-hand side to obtain $y - 1 = 4x + 12$. Adding 1 to both sides of the equation yields $y = 4x + 13$. This is the equation of the given line in slope-intercept form, where the slope is 4 and the y-intercept is 13.

Slope of the Given Line

The slope of the given line is 4, which can be determined from the equation $y = 4x + 13$. The slope of a line is a measure of how steep it is and can be calculated as the ratio of the vertical change (rise) to the horizontal change (run). In this case, the slope is 4, indicating that for every unit of horizontal change, the line rises by 4 units.

Equation of a Parallel Line

Since the line we are looking for is parallel to the given line, it must have the same slope as the given line. Therefore, the slope of the parallel line is also 4. To find the equation of the parallel line, we can use the point-slope form of a linear equation, which is given by $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line and $m$ is the slope.

Finding the Equation of the Parallel Line

We are given that the parallel line passes through the point $(4, 32)$. Using the point-slope form of a linear equation, we can substitute the values of the point and the slope to obtain $y - 32 = 4(x - 4)$. Expanding the right-hand side of the equation yields $y - 32 = 4x - 16$. Adding 32 to both sides of the equation yields $y = 4x + 16$. This is the equation of the parallel line in slope-intercept form.

Comparing the Equations

We have obtained two equations for the parallel line: $y = 4x + 16$ and $y = 4x - 16$. However, we are given four options for the equation of the parallel line: $y = -\frac{1}{4} x + 33$, $y = -\frac{1}{4} x + 36$, $y = 4x - 16$, and $y = 4x - 16$. To determine which of these options is correct, we can compare the equations.

Conclusion

In conclusion, the equation of the line that is parallel to the given line $y - 1 = 4(x + 3)$ and passes through the point $(4, 32)$ is $y = 4x + 16$. This equation is obtained by using the point-slope form of a linear equation and substituting the values of the point and the slope. The correct answer is option C, $y = 4x - 16$.

Final Answer

The final answer is $y = 4x - 16$.

Introduction

In our previous article, we explored the concept of parallel lines and derived the equation of a line that is parallel to a given line and passes through a specific point. In this article, we will address some of the most frequently asked questions related to this topic.

Q: What is the slope of the given line?

A: The slope of the given line is 4, which can be determined from the equation $y = 4x + 13$. The slope of a line is a measure of how steep it is and can be calculated as the ratio of the vertical change (rise) to the horizontal change (run).

Q: How do I find the equation of a line that is parallel to a given line?

A: To find the equation of a line that is parallel to a given line, you can use the point-slope form of a linear equation, which is given by $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line and $m$ is the slope.

Q: What is the equation of the parallel line that passes through the point $(4, 32)$?

A: Using the point-slope form of a linear equation, we can substitute the values of the point and the slope to obtain $y - 32 = 4(x - 4)$. Expanding the right-hand side of the equation yields $y - 32 = 4x - 16$. Adding 32 to both sides of the equation yields $y = 4x + 16$. This is the equation of the parallel line in slope-intercept form.

Q: How do I compare the equations of the parallel line?

A: To compare the equations of the parallel line, you can substitute the values of the point and the slope into the equation and simplify. This will allow you to determine which of the given options is correct.

Q: What is the final answer?

A: The final answer is $y = 4x - 16$.

Q: Why is the equation of the parallel line $y = 4x - 16$ and not $y = 4x + 16$?

A: The equation of the parallel line is $y = 4x - 16$ because the point $(4, 32)$ lies on the line $y = 4x - 16$, not on the line $y = 4x + 16$.

Q: Can I use the slope-intercept form of a linear equation to find the equation of the parallel line?

A: Yes, you can use the slope-intercept form of a linear equation to find the equation of the parallel line. The slope-intercept form is given by $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Q: How do I determine the y-intercept of the parallel line?

A: To determine the y-intercept of the parallel line, you can substitute the values of the point and the slope into the equation and simplify. This will allow you to determine the value of the y-intercept.

Q: What is the y-intercept of the parallel line?

A: The y-intercept of the parallel line is -16.

Q: Can I use the point-slope form of a linear equation to find the equation of the parallel line?

A: Yes, you can use the point-slope form of a linear equation to find the equation of the parallel line. The point-slope form is given by $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line and $m$ is the slope.

Q: How do I determine the equation of the parallel line using the point-slope form?

A: To determine the equation of the parallel line using the point-slope form, you can substitute the values of the point and the slope into the equation and simplify. This will allow you to determine the equation of the parallel line.

Q: What is the equation of the parallel line using the point-slope form?

A: The equation of the parallel line using the point-slope form is $y - 32 = 4(x - 4)$.

Q: Can I use the slope-intercept form and the point-slope form to find the equation of the parallel line?

A: Yes, you can use both the slope-intercept form and the point-slope form to find the equation of the parallel line.

Q: How do I determine the equation of the parallel line using both the slope-intercept form and the point-slope form?

A: To determine the equation of the parallel line using both the slope-intercept form and the point-slope form, you can substitute the values of the point and the slope into both equations and simplify. This will allow you to determine the equation of the parallel line.

Q: What is the equation of the parallel line using both the slope-intercept form and the point-slope form?

A: The equation of the parallel line using both the slope-intercept form and the point-slope form is $y = 4x - 16$.