If A Line Contains The Point (0, -1) And Has A Slope Of 2, Then Which Of The Following Points Also Lies On The Line?A. (1, 1) B. (0, 1) C. (2, 1)

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If a Line Contains the Point (0, -1) and Has a Slope of 2, Then Which of the Following Points Also Lies on the Line?

Understanding the Problem

The problem involves finding a point that lies on a line with a given slope and a known point on the line. The given information includes a point (0, -1) and a slope of 2. We need to determine which of the given points also lies on the line.

Recalling the Slope Formula

The slope of a line is calculated using the formula:

m = (y2 - y1) / (x2 - x1)

where m is the slope, and (x1, y1) and (x2, y2) are two points on the line.

Using the Given Point and Slope

We are given a point (0, -1) and a slope of 2. We can use the slope formula to find the equation of the line. However, since we are given the slope and a point, we can use the point-slope form of a line:

y - y1 = m(x - x1)

where (x1, y1) is the given point, and m is the slope.

Substituting the Given Values

Substituting the given values into the point-slope form, we get:

y - (-1) = 2(x - 0)

Simplifying the equation, we get:

y + 1 = 2x

Finding the Equation of the Line

The equation of the line is y + 1 = 2x. We can rewrite this equation in the slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.

y = 2x - 1

Evaluating the Given Points

We need to determine which of the given points (1, 1), (0, 1), and (2, 1) lies on the line. We can substitute each point into the equation of the line to see if it satisfies the equation.

Point (1, 1)

Substituting the point (1, 1) into the equation of the line, we get:

1 = 2(1) - 1

Simplifying the equation, we get:

1 = 2 - 1

1 = 1

The point (1, 1) satisfies the equation of the line.

Point (0, 1)

Substituting the point (0, 1) into the equation of the line, we get:

1 = 2(0) - 1

Simplifying the equation, we get:

1 = -1

The point (0, 1) does not satisfy the equation of the line.

Point (2, 1)

Substituting the point (2, 1) into the equation of the line, we get:

1 = 2(2) - 1

Simplifying the equation, we get:

1 = 4 - 1

1 = 3

The point (2, 1) does not satisfy the equation of the line.

Conclusion

Based on the equation of the line and the given points, we can conclude that the point (1, 1) lies on the line. The other two points, (0, 1) and (2, 1), do not lie on the line.

Key Takeaways

  • The slope of a line can be used to find the equation of the line.
  • The point-slope form of a line can be used to find the equation of the line given a point and the slope.
  • The equation of a line can be used to determine which points lie on the line.

Real-World Applications

  • The concept of slope and the equation of a line has many real-world applications, such as:
    • Calculating the steepness of a roof or a hill.
    • Determining the trajectory of a projectile.
    • Finding the equation of a road or a path.

Common Mistakes

  • One common mistake is to confuse the slope with the y-intercept.
  • Another common mistake is to substitute the wrong values into the equation of the line.

Tips and Tricks

  • To find the equation of a line, use the point-slope form and substitute the given values.
  • To determine which points lie on the line, substitute the points into the equation of the line and check if it satisfies the equation.

Conclusion

In conclusion, the point (1, 1) lies on the line with a slope of 2 and a point (0, -1). The other two points, (0, 1) and (2, 1), do not lie on the line. The concept of slope and the equation of a line has many real-world applications, and it is essential to understand the point-slope form and the equation of a line to solve problems involving lines.
Q&A: If a Line Contains the Point (0, -1) and Has a Slope of 2, Then Which of the Following Points Also Lies on the Line?

Frequently Asked Questions

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 using the formula m = (y2 - y1) / (x2 - x1), where m is the slope, and (x1, y1) and (x2, y2) are two points on the line.

Q: How do I find the equation of a line given a point and the slope?

A: You can use the point-slope form of a line, which is y - y1 = m(x - x1), where (x1, y1) is the given point, and m is the slope.

Q: What is the point-slope form of a line?

A: The point-slope form of a line is y - y1 = m(x - x1), where (x1, y1) is the given point, and m is the slope.

Q: How do I determine which points lie on the line?

A: You can substitute the points into the equation of the line and check if it satisfies the equation.

Q: What is the equation of the line with a slope of 2 and a point (0, -1)?

A: The equation of the line is y + 1 = 2x, which can be rewritten as y = 2x - 1.

Q: Which of the following points lies on the line with a slope of 2 and a point (0, -1)?

A: The point (1, 1) lies on the line.

Q: Why does the point (0, 1) not lie on the line?

A: The point (0, 1) does not lie on the line because it does not satisfy the equation of the line.

Q: Why does the point (2, 1) not lie on the line?

A: The point (2, 1) does not lie on the line because it does not satisfy the equation of the line.

Q: What are some real-world applications of the concept of slope and the equation of a line?

A: Some real-world applications of the concept of slope and the equation of a line include calculating the steepness of a roof or a hill, determining the trajectory of a projectile, and finding the equation of a road or a path.

Q: What are some common mistakes to avoid when working with the concept of slope and the equation of a line?

A: Some common mistakes to avoid include confusing the slope with the y-intercept and substituting the wrong values into the equation of the line.

Q: What are some tips and tricks for working with the concept of slope and the equation of a line?

A: Some tips and tricks for working with the concept of slope and the equation of a line include using the point-slope form and substituting the given values into the equation of the line.

Conclusion

In conclusion, the concept of slope and the equation of a line is a fundamental concept in mathematics that has many real-world applications. By understanding the point-slope form and the equation of a line, you can solve problems involving lines and determine which points lie on the line.