Which Equation Describes The Same Line As Y − 3 = − 1 ( X + 5 Y - 3 = -1(x + 5 Y − 3 = − 1 ( X + 5 ]?A. Y = − 1 X − 1 Y = -1x - 1 Y = − 1 X − 1 B. Y = − 1 X − 5 Y = -1x - 5 Y = − 1 X − 5 C. Y = − 1 X − 2 Y = -1x - 2 Y = − 1 X − 2 D. Y = − 1 X + 8 Y = -1x + 8 Y = − 1 X + 8

Which Equation Describes the Same Line as $y - 3 = -1(x + 5)$?

Understanding the Given Equation

The given equation is $y - 3 = -1(x + 5)$. To understand this equation, we need to simplify it and rewrite it in the slope-intercept form, which is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Simplifying the Given Equation

To simplify the given equation, we need to distribute the $-1$ to the terms inside the parentheses:

$$y - 3 = -1x - 5$$

Next, we can add $3$ to both sides of the equation to isolate $y$:

$$y = -1x - 5 + 3$$

Simplifying further, we get:

$$y = -1x - 2$$

Comparing with the Options

Now that we have simplified the given equation, we can compare it with the options provided:

A. $y = -1x - 1$ B. $y = -1x - 5$ C. $y = -1x - 2$ D. $y = -1x + 8$

From the simplified equation, we can see that the correct answer is:

C. $y = -1x - 2$

This is because the simplified equation and option C have the same slope and y-intercept.

Why the Other Options are Incorrect

Let's analyze why the other options are incorrect:

A. $y = -1x - 1$ has a different y-intercept, which means it represents a different line.

B. $y = -1x - 5$ has the same slope but a different y-intercept, which means it represents a different line.

D. $y = -1x + 8$ has a different slope and y-intercept, which means it represents a different line.

Conclusion

In conclusion, the equation that describes the same line as $y - 3 = -1(x + 5)$ is $y = -1x - 2$. This is because they have the same slope and y-intercept.

Understanding Slope and Y-Intercept

To understand why the equation $y = -1x - 2$ describes the same line as $y - 3 = -1(x + 5)$, we need to understand the concept of slope and y-intercept.

Slope

The slope of a line is a measure of how steep it is. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run). In the equation $y = -1x - 2$, the slope is $-1$, which means that for every unit we move to the right, we move down by one unit.

Y-Intercept

The y-intercept of a line is the point where the line intersects the y-axis. In the equation $y = -1x - 2$, the y-intercept is $-2$, which means that the line intersects the y-axis at the point $(0, -2)$.

Why Slope and Y-Intercept Matter

Slope and y-intercept matter because they determine the shape and position of a line. If two equations have the same slope and y-intercept, they represent the same line.

Real-World Applications

Understanding slope and y-intercept has many real-world applications. For example, in physics, the slope of a line can represent the acceleration of an object, while the y-intercept can represent the initial velocity.

Conclusion

In conclusion, the equation that describes the same line as $y - 3 = -1(x + 5)$ is $y = -1x - 2$. This is because they have the same slope and y-intercept. Understanding slope and y-intercept is crucial in mathematics and has many real-world applications.

Additional Tips and Tricks

Here are some additional tips and tricks to help you understand and work with equations:

• Simplify equations: Simplifying equations can help you identify the slope and y-intercept.
• Use the slope-intercept form: The slope-intercept form is a useful way to represent equations and identify the slope and y-intercept.
• Compare equations: Comparing equations can help you identify the slope and y-intercept.
• Use real-world examples: Using real-world examples can help you understand and apply the concept of slope and y-intercept.

Final Thoughts

In conclusion, the equation that describes the same line as $y - 3 = -1(x + 5)$ is $y = -1x - 2$. This is because they have the same slope and y-intercept. Understanding slope and y-intercept is crucial in mathematics and has many real-world applications.
Q&A: Understanding Equations and Slope

Q: What is the slope-intercept form of an equation?

A: The slope-intercept form of an equation is a way to represent an equation in the form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Q: What is the slope of a line?

A: The slope of a line is a measure of how steep it is. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run).

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

A: The y-intercept of a line is the point where the line intersects the y-axis.

Q: How do I determine the slope and y-intercept of an equation?

A: To determine the slope and y-intercept of an equation, you can simplify the equation and rewrite it in the slope-intercept form.

Q: What is the difference between a slope of 1 and a slope of -1?

A: A slope of 1 means that for every unit we move to the right, we move up by one unit. A slope of -1 means that for every unit we move to the right, we move down by one unit.

Q: How do I compare two equations to determine if they represent the same line?

A: To compare two equations, you can simplify them and rewrite them in the slope-intercept form. If the two equations have the same slope and y-intercept, they represent the same line.

Q: What are some real-world applications of understanding slope and y-intercept?

A: Understanding slope and y-intercept has many real-world applications, such as:

• Physics: The slope of a line can represent the acceleration of an object, while the y-intercept can represent the initial velocity.
• Engineering: The slope and y-intercept of a line can be used to design and build structures such as bridges and buildings.
• Economics: The slope and y-intercept of a line can be used to model and analyze economic data.

Q: How can I practice and improve my understanding of slope and y-intercept?

A: You can practice and improve your understanding of slope and y-intercept by:

• Simplifying and rewriting equations in the slope-intercept form.
• Comparing equations to determine if they represent the same line.
• Using real-world examples to apply the concept of slope and y-intercept.
• Working with different types of equations, such as linear and quadratic equations.

Q: What are some common mistakes to avoid when working with slope and y-intercept?

A: Some common mistakes to avoid when working with slope and y-intercept include:

• Not simplifying equations before rewriting them in the slope-intercept form.
• Not comparing equations to determine if they represent the same line.
• Not using real-world examples to apply the concept of slope and y-intercept.
• Not practicing and reviewing the concept of slope and y-intercept regularly.

Q: How can I use technology to help me understand and work with slope and y-intercept?

A: You can use technology such as graphing calculators and computer software to help you understand and work with slope and y-intercept. These tools can be used to:

• Graph equations and visualize the slope and y-intercept.
• Simplify and rewrite equations in the slope-intercept form.
• Compare equations to determine if they represent the same line.
• Work with different types of equations, such as linear and quadratic equations.

Q: What are some additional resources I can use to learn more about slope and y-intercept?

A: Some additional resources you can use to learn more about slope and y-intercept include:

• Textbooks and online resources that provide detailed explanations and examples.
• Video tutorials and online courses that provide step-by-step instructions and practice problems.
• Online communities and forums where you can ask questions and get help from others.
• Real-world examples and applications that demonstrate the importance and relevance of slope and y-intercept.