Determine The Intercepts Of The Line. Do Not Round Your Answers. \[ Y-3=5(x-2)\]

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Understanding the Problem

In this problem, we are given a linear equation in the form of a slope-intercept form, which is y - 3 = 5(x - 2). Our goal is to determine the intercepts of the line, which are the points where the line intersects the x-axis and the y-axis.

What are Intercepts?

Intercepts are the points where a line intersects the x-axis and the y-axis. The x-intercept is the point where the line intersects the x-axis, and the y-intercept is the point where the line intersects the y-axis.

Finding the Y-Intercept

To find the y-intercept, we need to find the value of y when x is equal to 0. In the given equation, we can substitute x = 0 and solve for y.

y - 3 = 5(x - 2) y - 3 = 5(0 - 2) y - 3 = 5(-2) y - 3 = -10 y = -10 + 3 y = -7

So, the y-intercept is (-7, 0).

Finding the X-Intercept

To find the x-intercept, we need to find the value of x when y is equal to 0. In the given equation, we can substitute y = 0 and solve for x.

y - 3 = 5(x - 2) 0 - 3 = 5(x - 2) -3 = 5(x - 2) -3 = 5x - 10 5x = -3 + 10 5x = 7 x = 7/5

So, the x-intercept is (7/5, 0).

Conclusion

In this problem, we determined the intercepts of the line by finding the y-intercept and the x-intercept. The y-intercept is (-7, 0) and the x-intercept is (7/5, 0).

Key Takeaways

  • The y-intercept is the point where the line intersects the y-axis.
  • The x-intercept is the point where the line intersects the x-axis.
  • To find the y-intercept, substitute x = 0 into the equation and solve for y.
  • To find the x-intercept, substitute y = 0 into the equation and solve for x.

Real-World Applications

Understanding intercepts is crucial in various real-world applications, such as:

  • Physics: Intercepts are used to describe the motion of objects, including the position and velocity of objects.
  • Engineering: Intercepts are used to design and optimize systems, including electrical circuits and mechanical systems.
  • Economics: Intercepts are used to analyze and model economic systems, including the demand and supply of goods and services.

Common Mistakes

  • Rounding answers: When solving for intercepts, it's essential to avoid rounding answers, as this can lead to inaccurate results.
  • Not following the order of operations: When solving for intercepts, it's essential to follow the order of operations (PEMDAS) to ensure accurate results.

Practice Problems

  1. Find the intercepts of the line y - 2 = 3(x - 1).
  2. Find the intercepts of the line y + 1 = 2(x + 3).
  3. Find the intercepts of the line y - 4 = x - 2.

Solutions

  1. y - 2 = 3(x - 1) y - 2 = 3x - 3 y = 3x - 1 y-intercept: (0, -1) x-intercept: (1/3, 0)
  2. y + 1 = 2(x + 3) y + 1 = 2x + 6 y = 2x + 5 y-intercept: (0, 5) x-intercept: (-5/2, 0)
  3. y - 4 = x - 2 y - 4 = x - 2 y = x - 2 + 4 y = x + 2 y-intercept: (0, 2) x-intercept: (-2, 0)

Conclusion

Q: What are intercepts in the context of a linear equation?

A: Intercepts are the points where a line intersects the x-axis and the y-axis. The x-intercept is the point where the line intersects the x-axis, and the y-intercept is the point where the line intersects the y-axis.

Q: How do I find the y-intercept of a linear equation?

A: To find the y-intercept, substitute x = 0 into the equation and solve for y. This will give you the value of y when x is equal to 0, which is the y-intercept.

Q: How do I find the x-intercept of a linear equation?

A: To find the x-intercept, substitute y = 0 into the equation and solve for x. This will give you the value of x when y is equal to 0, which is the x-intercept.

Q: What is the difference between the y-intercept and the x-intercept?

A: The y-intercept is the point where the line intersects the y-axis, and the x-intercept is the point where the line intersects the x-axis. The y-intercept is the value of y when x is equal to 0, and the x-intercept is the value of x when y is equal to 0.

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

A: Yes, you can use the slope-intercept form of a linear equation to find the intercepts. The slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept. To find the x-intercept, substitute y = 0 into the equation and solve for x.

Q: What are some common mistakes to avoid when finding intercepts?

A: Some common mistakes to avoid when finding intercepts include:

  • Rounding answers: When solving for intercepts, it's essential to avoid rounding answers, as this can lead to inaccurate results.
  • Not following the order of operations: When solving for intercepts, it's essential to follow the order of operations (PEMDAS) to ensure accurate results.

Q: How do I apply the concept of intercepts in real-world scenarios?

A: Understanding intercepts is crucial in various real-world applications, such as:

  • Physics: Intercepts are used to describe the motion of objects, including the position and velocity of objects.
  • Engineering: Intercepts are used to design and optimize systems, including electrical circuits and mechanical systems.
  • Economics: Intercepts are used to analyze and model economic systems, including the demand and supply of goods and services.

Q: Can I use the concept of intercepts to solve systems of linear equations?

A: Yes, you can use the concept of intercepts to solve systems of linear equations. By finding the intercepts of each equation, you can determine the point of intersection between the two lines.

Q: What are some practice problems to help reinforce the concept of intercepts?

A: Here are some practice problems to help reinforce the concept of intercepts:

  1. Find the intercepts of the line y - 2 = 3(x - 1).
  2. Find the intercepts of the line y + 1 = 2(x + 3).
  3. Find the intercepts of the line y - 4 = x - 2.

Solutions

  1. y - 2 = 3(x - 1) y - 2 = 3x - 3 y = 3x - 1 y-intercept: (0, -1) x-intercept: (1/3, 0)
  2. y + 1 = 2(x + 3) y + 1 = 2x + 6 y = 2x + 5 y-intercept: (0, 5) x-intercept: (-5/2, 0)
  3. y - 4 = x - 2 y - 4 = x - 2 y = x - 2 + 4 y = x + 2 y-intercept: (0, 2) x-intercept: (-2, 0)

Conclusion

In this Q&A article, we covered various questions and answers related to the concept of intercepts in the context of linear equations. We discussed how to find the y-intercept and the x-intercept, and how to apply the concept of intercepts in real-world scenarios. We also provided practice problems to help reinforce the concept of intercepts.