A Line Has A Slope Of 4 And A $y$-intercept Of $\frac{3}{5}$. Write Its Equation In Slope-intercept Form.Write Your Answer Using Integers, Proper Fractions, And Improper Fractions In Simplest Form.

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Introduction

In mathematics, the slope-intercept form of a line is a fundamental concept that helps us understand the relationship between the slope and the y-intercept of a line. The slope-intercept form of a line is given by the equation y = mx + b, where m is the slope of the line and b is the y-intercept. In this article, we will explore how to write the equation of a line in slope-intercept form when given its slope and y-intercept.

Understanding the Slope-Intercept Form

The slope-intercept form of a line is a powerful tool that allows us to visualize the relationship between the slope and the y-intercept of a line. The slope of a line is a measure of how steep it is, while the y-intercept is the point at which the line intersects the y-axis. The slope-intercept form of a line is given by the equation y = mx + b, where m is the slope of the line and b is the y-intercept.

Slope (m)

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) between two points on the line. The slope of a line can be positive, negative, or zero, depending on the direction and steepness of the line.

Y-Intercept (b)

The y-intercept of a line is the point at which the line intersects the y-axis. It is the value of y when x is equal to zero. The y-intercept can be a positive or negative value, depending on the position of the line relative to the y-axis.

Writing the Equation of a Line in Slope-Intercept Form

Now that we have a good understanding of the slope-intercept form of a line, let's see how to write the equation of a line in slope-intercept form when given its slope and y-intercept.

Example 1: Slope = 4, Y-Intercept = 3/5

Let's say we are given a line with a slope of 4 and a y-intercept of 3/5. We can write the equation of this line in slope-intercept form as follows:

y = 4x + 3/5

In this equation, the slope (m) is 4, and the y-intercept (b) is 3/5.

Example 2: Slope = -2, Y-Intercept = 1/2

Let's say we are given a line with a slope of -2 and a y-intercept of 1/2. We can write the equation of this line in slope-intercept form as follows:

y = -2x + 1/2

In this equation, the slope (m) is -2, and the y-intercept (b) is 1/2.

Conclusion

In conclusion, writing the equation of a line in slope-intercept form is a straightforward process that involves substituting the given slope and y-intercept into the equation y = mx + b. By following the steps outlined in this article, you should be able to write the equation of a line in slope-intercept form when given its slope and y-intercept.

Frequently Asked Questions

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

A: The slope-intercept form of a line is given by the equation y = mx + b, where m is the slope of the line and b is the y-intercept.

Q: How do I write the equation of a line in slope-intercept form?

A: To write the equation of a line in slope-intercept form, substitute the given slope and y-intercept into the equation y = mx + b.

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

A: The y-intercept of a line is the point at which the line intersects the y-axis. It is the value of y when x is equal to zero.

Q: Can the y-intercept be a negative value?

A: Yes, the y-intercept can be a negative value, depending on the position of the line relative to the y-axis.

References

Further Reading

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Q&A: Writing the Equation of a Line in Slope-Intercept Form

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

A: The slope-intercept form of a line is given by the equation y = mx + b, where m is the slope of the line and b is the y-intercept.

Q: How do I write the equation of a line in slope-intercept form?

A: To write the equation of a line in slope-intercept form, substitute the given slope and y-intercept into the equation y = mx + b.

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

A: The y-intercept of a line is the point at which the line intersects the y-axis. It is the value of y when x is equal to zero.

Q: Can the y-intercept be a negative value?

A: Yes, the y-intercept can be a negative value, depending on the position of the line relative to the y-axis.

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) between two points on the line.

Q: Can the slope be a negative value?

A: Yes, the slope can be a negative value, depending on the direction of the line.

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

A: To find the slope of a line, use the formula m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are two points on the line.

Q: What is the equation of a horizontal line?

A: The equation of a horizontal line is y = b, where b is the y-intercept.

Q: What is the equation of a vertical line?

A: The equation of a vertical line is x = a, where a is the x-intercept.

Q: Can a line have a slope of zero?

A: Yes, a line can have a slope of zero, which means it is a horizontal line.

Q: Can a line have a slope of infinity?

A: No, a line cannot have a slope of infinity, but it can have a very large slope.

Q: How do I graph a line in slope-intercept form?

A: To graph a line in slope-intercept form, plot the y-intercept and then use the slope to find another point on the line.

Q: What is the significance of the slope-intercept form?

A: The slope-intercept form is significant because it allows us to easily identify the slope and y-intercept of a line, which are essential components of the line's equation.

Real-World Applications of Slope-Intercept Form

The slope-intercept form has numerous real-world applications, including:

  • Physics: The slope-intercept form is used to describe the motion of objects under the influence of gravity.
  • Engineering: The slope-intercept form is used to design and optimize systems, such as bridges and buildings.
  • Economics: The slope-intercept form is used to model economic systems and make predictions about future trends.
  • Computer Science: The slope-intercept form is used in computer graphics and game development to create realistic and interactive environments.

Conclusion

In conclusion, the slope-intercept form is a powerful tool that allows us to easily identify the slope and y-intercept of a line. By understanding the slope-intercept form, we can apply it to various real-world applications and make predictions about future trends.

Frequently Asked Questions

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

A: The slope-intercept form of a line is given by the equation y = mx + b, where m is the slope of the line and b is the y-intercept.

Q: How do I write the equation of a line in slope-intercept form?

A: To write the equation of a line in slope-intercept form, substitute the given slope and y-intercept into the equation y = mx + b.

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

A: The y-intercept of a line is the point at which the line intersects the y-axis. It is the value of y when x is equal to zero.

Q: Can the y-intercept be a negative value?

A: Yes, the y-intercept can be a negative value, depending on the position of the line relative to the y-axis.

References

Further Reading