Determine The Equation Of A Line That Passes Through The Point (4, 7) And Has A Slope Of 3.

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Introduction

In mathematics, the equation of a line is a fundamental concept that is used to describe the relationship between two variables. The equation of a line can be expressed in various forms, including the slope-intercept form, point-slope form, and standard form. In this article, we will focus on determining the equation of a line that passes through a given point and has a specified slope.

Understanding the Slope-Intercept Form

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. The slope of a line is a measure of how steep it is, and it is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. The y-intercept is the point where the line intersects the y-axis.

Understanding the Point-Slope Form

The point-slope form of a line is given by the equation y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope of the line. This form is useful when we know the slope of the line and a point that it passes through.

Determining the Equation of a Line

To determine the equation of a line that passes through a given point and has a specified slope, we can use the point-slope form of the equation. Let's consider the example of a line that passes through the point (4, 7) and has a slope of 3.

Step 1: Write Down the Point-Slope Form

The point-slope form of the equation is y - y1 = m(x - x1). In this case, the point (x1, y1) is (4, 7) and the slope m is 3. So, the equation becomes:

y - 7 = 3(x - 4)

Step 2: Simplify the Equation

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

y - 7 = 3x - 12

Step 3: Add 7 to Both Sides

To isolate y, we can add 7 to both sides of the equation:

y = 3x - 12 + 7

Step 4: Simplify the Right-Hand Side

To simplify the right-hand side, we can combine the constants:

y = 3x - 5

Step 5: Write the Equation in Slope-Intercept Form

To write the equation in slope-intercept form, we can rearrange the terms:

y = 3x - 5

This is the equation of the line in slope-intercept form.

Conclusion

In this article, we have determined the equation of a line that passes through the point (4, 7) and has a slope of 3. We have used the point-slope form of the equation and simplified it to obtain the equation in slope-intercept form. This is a fundamental concept in mathematics, and it has many practical applications in fields such as physics, engineering, and economics.

Real-World Applications

The equation of a line has many real-world applications. For example, it can be used to model the relationship between two variables, such as the cost of a product and its quantity sold. It can also be used to determine the equation of a line that passes through a given point and has a specified slope.

Tips and Tricks

Here are some tips and tricks for determining the equation of a line:

  • Use the point-slope form: The point-slope form is a useful tool for determining the equation of a line that passes through a given point and has a specified slope.
  • Simplify the equation: Simplifying the equation can make it easier to read and understand.
  • Use the slope-intercept form: The slope-intercept form is a useful tool for determining the equation of a line in a more compact form.

Common Mistakes

Here are some common mistakes to avoid when determining the equation of a line:

  • Not using the point-slope form: Failing to use the point-slope form can make it difficult to determine the equation of a line.
  • Not simplifying the equation: Failing to simplify the equation can make it difficult to read and understand.
  • Not using the slope-intercept form: Failing to use the slope-intercept form can make it difficult to determine the equation of a line in a more compact form.

Conclusion

In conclusion, determining the equation of a line is a fundamental concept in mathematics that has many practical applications. By using the point-slope form and simplifying the equation, we can determine the equation of a line that passes through a given point and has a specified slope. This is a useful tool for modeling the relationship between two variables and has many real-world applications.

Introduction

Determining the equation of a line is a fundamental concept in mathematics that has many practical applications. In our previous article, we discussed how to determine the equation of a line that passes through a given point and has a specified slope. In this article, we will answer some frequently asked questions about determining the equation of a line.

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

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

Q: How do I determine the equation of a line that passes through a given point and has a specified slope?

A: To determine the equation of a line that passes through a given point and has a specified slope, you can use the point-slope form of the equation. Simply substitute the given point and slope into the equation and simplify.

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 convert the point-slope form to the slope-intercept form?

A: To convert the point-slope form to the slope-intercept form, you can simplify the equation by combining like terms and isolating y.

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

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

Q: How do I determine the y-intercept of a line?

A: To determine the y-intercept of a line, you can substitute x = 0 into the equation of the line and solve for y.

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: How do I determine the slope of a line?

A: To determine the slope of a line, you can 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 given by the equation y = b, where b is the y-intercept of the line.

Q: What is the equation of a vertical line?

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

Q: How do I determine the equation of a line that passes through two points?

A: To determine the equation of a line that passes through two points, you can use the point-slope form of the equation and substitute the two points into the equation.

Q: What is the equation of a line that passes through the origin?

A: The equation of a line that passes through the origin is given by the equation y = mx, where m is the slope of the line.

Conclusion

In conclusion, determining the equation of a line is a fundamental concept in mathematics that has many practical applications. By understanding the point-slope form, slope-intercept form, and y-intercept of a line, you can determine the equation of a line that passes through a given point and has a specified slope. We hope this article has been helpful in answering your questions about determining the equation of a line.