Alejandra Correctly Wrote The Equation Y − 3 = 1 5 ( X − 10 Y - 3 = \frac{1}{5}(x - 10 Y − 3 = 5 1 ​ ( X − 10 ] To Represent A Line That Her Teacher Sketched. The Teacher Then Changed The Line So It Had A Slope Of 2, But Still Went Through The Same Point. Which Equation Should

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Introduction

In mathematics, a line can be represented by an equation in the form of y = mx + b, where m is the slope and b is the y-intercept. The equation provided by Alejandra, y - 3 = \frac{1}{5}(x - 10), represents a line with a slope of \frac{1}{5} and a y-intercept of 3. However, the teacher changed the line to have a slope of 2, but still went through the same point. In this article, we will explore the new equation that represents the line with a slope of 2.

The Original Equation

The original equation provided by Alejandra is y - 3 = \frac{1}{5}(x - 10). To find the new equation with a slope of 2, we need to understand the relationship between the original equation and the new equation.

Finding the Point of Intersection

To find the new equation, we need to find the point of intersection between the original line and the new line. The point of intersection is the point where the two lines meet. Since the new line has a slope of 2, we can use the point-slope form of a line to find the equation.

Using the Point-Slope Form

The point-slope form of a line is given by y - y1 = m(x - x1), where (x1, y1) is the point of intersection and m is the slope. We know that the point of intersection is the same as the point on the original line, which is (10, 3). We also know that the slope of the new line is 2.

Finding the New Equation

Using the point-slope form, we can find the new equation by substituting the values of the point of intersection and the slope.

y - 3 = 2(x - 10)

Simplifying the Equation

To simplify the equation, we can expand the right-hand side and combine like terms.

y - 3 = 2x - 20

Adding 3 to Both Sides

To isolate y, we can add 3 to both sides of the equation.

y = 2x - 17

Conclusion

The new equation that represents the line with a slope of 2 is y = 2x - 17. This equation is different from the original equation, which had a slope of \frac{1}{5}. The new equation has a slope of 2 and still passes through the same point as the original line.

Understanding the Concept of Slope

The concept of slope is a fundamental concept in mathematics. The slope of a line represents the rate of change of the line. A line with a positive slope rises from left to right, while a line with a negative slope falls from left to right. A line with a slope of 0 is a horizontal line.

Types of Slopes

There are several types of slopes, including:

  • Positive Slope: A line with a positive slope rises from left to right.
  • Negative Slope: A line with a negative slope falls from left to right.
  • Zero Slope: A line with a slope of 0 is a horizontal line.
  • Undefined Slope: A line with an undefined slope is a vertical line.

Real-World Applications of Slope

The concept of slope has several real-world applications, including:

  • Physics: The slope of a line can represent the velocity of an object.
  • Engineering: The slope of a line can represent the angle of a ramp or a road.
  • Economics: The slope of a line can represent the rate of change of a variable.

Conclusion

In conclusion, the new equation that represents the line with a slope of 2 is y = 2x - 17. This equation is different from the original equation, which had a slope of \frac{1}{5}. The new equation has a slope of 2 and still passes through the same point as the original line. The concept of slope is a fundamental concept in mathematics, and it has several real-world applications.

References

  • Mathematics Textbook: A textbook on mathematics that covers the concept of slope.
  • Online Resources: Online resources that provide information on the concept of slope.

Further Reading

  • Slope and Graphing: A article that provides information on the concept of slope and graphing.
  • Mathematics and Real-World Applications: An article that provides information on the real-world applications of mathematics.
    Q&A: Understanding the Concept of Slope =============================================

Introduction

In our previous article, we explored the concept of slope and how it can be used to represent the rate of change of a line. In this article, we will answer some frequently asked questions about the concept of slope.

Q: What is the slope of a line?

A: The slope of a line is a measure of how steep the line is. It is calculated by dividing the vertical change (rise) by the horizontal change (run).

Q: How do I calculate the slope of a line?

A: To calculate the slope of a line, you can use the following formula:

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

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

Q: What is the difference between a positive slope and a negative slope?

A: A positive slope represents a line that rises from left to right, while a negative slope represents a line that falls from left to right.

Q: What is the slope of a horizontal line?

A: The slope of a horizontal line is 0, since there is no vertical change.

Q: What is the slope of a vertical line?

A: The slope of a vertical line is undefined, since there is no horizontal change.

Q: How do I use the concept of slope in real-world applications?

A: The concept of slope has several real-world applications, including:

  • Physics: The slope of a line can represent the velocity of an object.
  • Engineering: The slope of a line can represent the angle of a ramp or a road.
  • Economics: The slope of a line can represent the rate of change of a variable.

Q: Can I use the concept of slope to solve problems in other areas of mathematics?

A: Yes, the concept of slope can be used to solve problems in other areas of mathematics, including:

  • Algebra: The concept of slope can be used to solve systems of linear equations.
  • Geometry: The concept of slope can be used to find the equation of a line that passes through two points.
  • Calculus: The concept of slope can be used to find the derivative of a function.

Q: How do I graph a line with a given slope?

A: To graph a line with a given slope, you can use the following steps:

  1. Choose a point on the line.
  2. Use the slope to find the equation of the line.
  3. Plot the line on a coordinate plane.

Q: Can I use a calculator to find the slope of a line?

A: Yes, you can use a calculator to find the slope of a line. Most calculators have a built-in function for calculating the slope of a line.

Conclusion

In conclusion, the concept of slope is a fundamental concept in mathematics that has several real-world applications. By understanding the concept of slope, you can solve problems in other areas of mathematics and apply the concept to real-world situations.

References

  • Mathematics Textbook: A textbook on mathematics that covers the concept of slope.
  • Online Resources: Online resources that provide information on the concept of slope.

Further Reading

  • Slope and Graphing: An article that provides information on the concept of slope and graphing.
  • Mathematics and Real-World Applications: An article that provides information on the real-world applications of mathematics.