A Line Has A Slope Of 3 And Passes Through The Point \[$(-2,-10)\$\]. 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 linear equation is a fundamental concept used to represent lines on a coordinate plane. The slope-intercept form is given by the equation y = mx + b, where m represents the slope of the line and b is the y-intercept. In this article, we will explore how to find the equation of a line with a given slope and a point it passes through, using the slope-intercept form.

Understanding the Slope-Intercept Form

The slope-intercept form of a linear equation is a powerful tool for representing lines on a coordinate plane. The equation y = mx + b consists of two main components: the slope (m) and the y-intercept (b). The slope (m) represents the rate of change of the line, while the y-intercept (b) represents the point at which the line intersects the y-axis.

Finding the Equation of a Line with a Given Slope and Point

To find the equation of a line with a given slope and a point it passes through, we can use the slope-intercept form of a linear equation. The given information includes the slope (m = 3) and a point the line passes through (x = -2, y = -10). We can use this information to find the y-intercept (b) of the line.

Using the Point-Slope Form to Find the Equation

The point-slope form of a linear equation is given by the equation y - y1 = m(x - x1), where (x1, y1) is a point the line passes through and m is the slope. We can use this form to find the equation of the line with a given slope and a point it passes through.

Substituting the Given Values into the Point-Slope Form

We are given the slope (m = 3) and a point the line passes through (x = -2, y = -10). We can substitute these values into the point-slope form of a linear equation:

y - (-10) = 3(x - (-2))

Simplifying the Equation

We can simplify the equation by combining like terms:

y + 10 = 3(x + 2)

Distributing the Slope

We can distribute the slope (3) to the terms inside the parentheses:

y + 10 = 3x + 6

Isolating the y-Term

We can isolate the y-term by subtracting 10 from both sides of the equation:

y = 3x - 4

Conclusion

In this article, we explored how to find the equation of a line with a given slope and a point it passes through, using the slope-intercept form. We used the point-slope form of a linear equation to find the equation of the line, and then simplified the equation to find the y-intercept. The final equation of the line is y = 3x - 4.

Example Use Case

Suppose we want to find the equation of a line with a slope of 2 and a point it passes through (x = 1, y = 3). We can use the point-slope form of a linear equation to find the equation of the line:

y - 3 = 2(x - 1)

Simplifying the equation, we get:

y - 3 = 2x - 2

Isolating the y-term, we get:

y = 2x + 1

This is the equation of the line with a slope of 2 and a point it passes through (x = 1, y = 3).

Applications of the Slope-Intercept Form

The slope-intercept form of a linear equation has many applications in mathematics and real-world scenarios. Some examples include:

  • Linear Regression: The slope-intercept form is used to model the relationship between two variables in linear regression.
  • Physics: The slope-intercept form is used to describe the motion of objects under constant acceleration.
  • Economics: The slope-intercept form is used to model the relationship between two variables in economics, such as the demand for a product.

Conclusion

In conclusion, the slope-intercept form of a linear equation is a powerful tool for representing lines on a coordinate plane. We can use the point-slope form of a linear equation to find the equation of a line with a given slope and a point it passes through. The final equation of the line is y = 3x - 4. The slope-intercept form has many applications in mathematics and real-world scenarios, including linear regression, physics, and economics.

Introduction

In our previous article, we explored how to find the equation of a line with a given slope and a point it passes through, using the slope-intercept form. In this article, we will answer some frequently asked questions (FAQs) related to the topic.

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

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

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

A: To find the equation of a line with a given slope and a point it passes through, you can use the point-slope form of a linear equation: y - y1 = m(x - x1), where (x1, y1) is a point the line passes through and m is the slope.

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

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

Q: How do I simplify the equation after using the point-slope form?

A: To simplify the equation after using the point-slope form, you can combine like terms and isolate the y-term.

Q: What is the final equation of the line with a slope of 3 and a point it passes through (x = -2, y = -10)?

A: The final equation of the line with a slope of 3 and a point it passes through (x = -2, y = -10) is y = 3x - 4.

Q: Can I use the slope-intercept form to find the equation of a line with a given slope and a point it passes through?

A: Yes, you can use the slope-intercept form to find the equation of a line with a given slope and a point it passes through. However, it is often easier to use the point-slope form.

Q: What are some applications of the slope-intercept form?

A: The slope-intercept form has many applications in mathematics and real-world scenarios, including linear regression, physics, and economics.

Q: Can I use the slope-intercept form to model the relationship between two variables in economics?

A: Yes, you can use the slope-intercept form to model the relationship between two variables in economics.

Q: What is the y-intercept in the equation y = 3x - 4?

A: The y-intercept in the equation y = 3x - 4 is -4.

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

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

Q: Can I use the slope-intercept form to find the equation of a line with a given slope and a point it passes through if the point is not on the y-axis?

A: Yes, you can use the slope-intercept form to find the equation of a line with a given slope and a point it passes through if the point is not on the y-axis.

Q: What is the equation of a line with a slope of 2 and a point it passes through (x = 1, y = 3)?

A: The equation of a line with a slope of 2 and a point it passes through (x = 1, y = 3) is y = 2x + 1.

Conclusion

In this article, we answered some frequently asked questions (FAQs) related to finding the equation of a line with a given slope and a point it passes through, using the slope-intercept form. We hope this article has been helpful in clarifying any doubts you may have had.