Determine The Slope-intercept Form Of The Linear Equation:$\frac{x}{2} - Y + 6 = 0$A. $y = \frac{x}{2} + 6$ B. $y = \frac{x}{2} - 6$ C. $y = \frac{x}{2} - 6$

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Introduction

In mathematics, the slope-intercept form of a linear equation is a fundamental concept that helps us understand the relationship between the variables in an equation. The slope-intercept form 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 determine the slope-intercept form of the linear equation x2−y+6=0\frac{x}{2} - y + 6 = 0.

Understanding the Given Equation

The given equation is x2−y+6=0\frac{x}{2} - y + 6 = 0. To determine the slope-intercept form, we need to isolate the variable y. We can start by subtracting x2\frac{x}{2} from both sides of the equation.

x2−y+6=0\frac{x}{2} - y + 6 = 0

Subtracting x2\frac{x}{2} from both sides:

−y+6=−x2-y + 6 = -\frac{x}{2}

Isolating the Variable y

Next, we need to isolate the variable y. We can do this by subtracting 6 from both sides of the equation.

−y+6=−x2-y + 6 = -\frac{x}{2}

Subtracting 6 from both sides:

−y=−x2−6-y = -\frac{x}{2} - 6

Multiplying Both Sides by -1

To make the equation more manageable, we can multiply both sides by -1.

−y=−x2−6-y = -\frac{x}{2} - 6

Multiplying both sides by -1:

y=x2+6y = \frac{x}{2} + 6

Conclusion

In conclusion, the slope-intercept form of the linear equation x2−y+6=0\frac{x}{2} - y + 6 = 0 is y=x2+6y = \frac{x}{2} + 6. This equation represents a linear relationship between the variables x and y, where the slope is 12\frac{1}{2} and the y-intercept is 6.

Comparison with the Given Options

Let's compare our result with the given options:

A. y=x2+6y = \frac{x}{2} + 6 B. y=x2−6y = \frac{x}{2} - 6 C. y=x2−6y = \frac{x}{2} - 6

Our result matches option A, which is y=x2+6y = \frac{x}{2} + 6.

Final Answer

The final answer is:

A. y=x2+6y = \frac{x}{2} + 6

Additional Information

The slope-intercept form of a linear equation is a powerful tool for understanding the relationship between the variables in an equation. By isolating the variable y, we can determine the slope and y-intercept of the line, which can be used to graph the line and solve for the variables.

Common Applications

The slope-intercept form of a linear equation has many practical applications in mathematics and science. For example, it can be used to model the relationship between two variables, such as the cost of a product and the quantity sold. It can also be used to determine the equation of a line given two points on the line.

Real-World Examples

Here are a few real-world examples of the slope-intercept form of a linear equation:

  • The cost of a product is $10 per unit, and the quantity sold is 20 units. The equation of the line is y = 10x + 200, where y is the total cost and x is the quantity sold.
  • The height of a ball is given by the equation y = -16t^2 + 100, where y is the height and t is the time in seconds.

Conclusion

Introduction

In our previous article, we determined the slope-intercept form of the linear equation x2−y+6=0\frac{x}{2} - y + 6 = 0. In this article, we will answer some frequently asked questions about the slope-intercept form of a linear equation.

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 is the slope of the line and b is the y-intercept.

Q: How do I determine the slope-intercept form of a linear equation?

A: To determine the slope-intercept form of a linear equation, you need to isolate the variable y. You can do this by subtracting the x-term from both sides of the equation and then adding or subtracting the constant term.

Q: What is the slope of a linear equation?

A: The slope of a linear equation is the coefficient of the x-term in the equation. In the equation y = mx + b, the slope is m.

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

A: The y-intercept of a linear equation is the constant term in the equation. In the equation y = mx + b, the y-intercept is b.

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

A: To graph a linear equation in slope-intercept form, you can use the slope and y-intercept to plot two points on the line. The slope tells you how steep the line is, and the y-intercept tells you where the line crosses the y-axis.

Q: Can I use the slope-intercept form to solve a system of linear equations?

A: Yes, you can use the slope-intercept form to solve a system of linear equations. By equating the two equations, you can solve for the variables.

Q: What are some common applications of the slope-intercept form of a linear equation?

A: The slope-intercept form of a linear equation has many practical applications in mathematics and science. It can be used to model the relationship between two variables, such as the cost of a product and the quantity sold. It can also be used to determine the equation of a line given two points on the line.

Q: Can I use the slope-intercept form to find the equation of a line given two points?

A: Yes, you can use the slope-intercept form to find the equation of a line given two points. By using the slope formula and the y-intercept formula, you can determine the equation of the line.

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

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

  • Not isolating the variable y
  • Not using the correct slope and y-intercept
  • Not graphing the line correctly
  • Not using the correct equation to solve a system of linear equations

Conclusion

In conclusion, the slope-intercept form of a linear equation is a powerful tool for understanding the relationship between the variables in an equation. By isolating the variable y, we can determine the slope and y-intercept of the line, which can be used to graph the line and solve for the variables. We hope this Q&A article has been helpful in answering some of your questions about the slope-intercept form of a linear equation.

Additional Resources

For more information on the slope-intercept form of a linear equation, we recommend the following resources:

  • Khan Academy: Slope-Intercept Form
  • Mathway: Slope-Intercept Form
  • Wolfram Alpha: Slope-Intercept Form

Final Answer

The final answer is:

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