The Point-slope Form Of The Equation Of A Line That Passes Through Points { (8,4)$}$ And { (0,2)$}$ Is { Y - 4 = \frac{1}{4}(x - 8)$} . W H A T I S T H E S L O P E − I N T E R C E P T F O R M O F T H E E Q U A T I O N F O R T H I S L I N E ? A . \[ . What Is The Slope-intercept Form Of The Equation For This Line?A. \[ . Wha T I S T H Es L O P E − In T Erce Pt F Or M O F T H Ee Q U A T I O N F Or T Hi S L In E ? A . \[ Y =

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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 is given by the equation y = mx + b, where m is the slope and b is the y-intercept. In this article, we will explore the slope-intercept form of a line and provide a step-by-step guide on how to convert the point-slope form to the slope-intercept form.

What is the Slope-Intercept Form?

The slope-intercept form of a line is a way to express the equation of a line in terms of its slope and y-intercept. The slope-intercept form is given by the equation y = mx + b, where m is the slope and b is the y-intercept. The slope-intercept form is a more intuitive and easier-to-use form of the equation of a line, as it allows us to easily identify the slope and y-intercept of the line.

Converting Point-Slope Form to Slope-Intercept Form

To convert the point-slope form to the slope-intercept form, we need to isolate the y-variable on one side of the equation. The point-slope form is given by the equation y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope. To convert this form to the slope-intercept form, we need to add y1 to both sides of the equation and then simplify.

Step 1: Add y1 to Both Sides of the Equation

The first step in converting the point-slope form to the slope-intercept form is to add y1 to both sides of the equation. This will give us the equation y = m(x - x1) + y1.

Step 2: Simplify the Equation

The next step is to simplify the equation by distributing the slope (m) to the terms inside the parentheses. This will give us the equation y = mx - mx1 + y1.

Step 3: Isolate the y-Variable

The final step is to isolate the y-variable on one side of the equation. To do this, we need to move the terms involving x to the other side of the equation. This will give us the equation y = mx + (y1 - mx1).

Example: Converting Point-Slope Form to Slope-Intercept Form

Let's consider an example to illustrate the process of converting the point-slope form to the slope-intercept form. The point-slope form of the equation of a line that passes through points (8, 4) and (0, 2) is given by the equation y - 4 = (1/4)(x - 8). To convert this form to the slope-intercept form, we need to follow the steps outlined above.

Step 1: Add 4 to Both Sides of the Equation

The first step is to add 4 to both sides of the equation. This will give us the equation y = (1/4)(x - 8) + 4.

Step 2: Simplify the Equation

The next step is to simplify the equation by distributing the slope (1/4) to the terms inside the parentheses. This will give us the equation y = (1/4)x - 2 + 4.

Step 3: Isolate the y-Variable

The final step is to isolate the y-variable on one side of the equation. To do this, we need to move the terms involving x to the other side of the equation. This will give us the equation y = (1/4)x + 2.

Conclusion

In conclusion, the slope-intercept form of a line is a fundamental concept in mathematics that helps us understand the relationship between the slope and the y-intercept of a line. The slope-intercept form is given by the equation y = mx + b, where m is the slope and b is the y-intercept. To convert the point-slope form to the slope-intercept form, we need to follow the steps outlined above. By following these steps, we can easily convert the point-slope form to the slope-intercept form and identify the slope and y-intercept of a line.

Final Answer

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

A: The slope-intercept form of a line is a way to express the equation of a line in terms of its slope and y-intercept. The slope-intercept form is given by the equation y = mx + b, where m is the slope 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 need to follow the steps outlined above. First, add the y-coordinate of the point to both sides of the equation. Then, simplify the equation by distributing the slope to the terms inside the parentheses. Finally, isolate the y-variable on one side of the equation.

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

A: The slope of a line in the slope-intercept form is the coefficient of the x-term, which is the number that is multiplied by x. In the equation y = mx + b, the slope is m.

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

A: The y-intercept of a line in the slope-intercept form is the constant term, which is the number that is added to the product of the slope and x. In the equation y = mx + b, the y-intercept is b.

Q: How do I find the slope and y-intercept of a line given its equation in the slope-intercept form?

A: To find the slope and y-intercept of a line given its equation in the slope-intercept form, you can simply look at the equation and identify the slope and y-intercept. In the equation y = mx + b, the slope is m and the y-intercept is b.

Q: Can I use the slope-intercept form to graph a line?

A: Yes, you can use the slope-intercept form to graph a line. To graph a line in the slope-intercept form, you can plot the y-intercept and then use the slope to find other points on the line.

Q: What are some common mistakes to avoid when working with the slope-intercept form?

A: Some common mistakes to avoid when working with the slope-intercept form include:

  • Not distributing the slope to the terms inside the parentheses when converting from point-slope form to slope-intercept form.
  • Not isolating the y-variable on one side of the equation when converting from point-slope form to slope-intercept form.
  • Not identifying the slope and y-intercept correctly when given the equation in the slope-intercept form.

Q: How do I use the slope-intercept form to solve problems involving lines?

A: To use the slope-intercept form to solve problems involving lines, you can follow these steps:

  • Identify the slope and y-intercept of the line.
  • Use the slope and y-intercept to find other points on the line.
  • Use the points on the line to graph the line.
  • Use the line to solve problems involving lines, such as finding the equation of a line that passes through two points.

Conclusion

In conclusion, the slope-intercept form of a line is a fundamental concept in mathematics that helps us understand the relationship between the slope and the y-intercept of a line. By following the steps outlined above, you can convert the point-slope form to the slope-intercept form and identify the slope and y-intercept of a line. With practice and experience, you can use the slope-intercept form to solve problems involving lines and become proficient in graphing and solving equations of lines.