What Is The \[$ Y \$\]-intercept Of The Line \[$ 4y + 7 = 2x - 5 \$\]?

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What is the { y $}$-intercept of the line { 4y + 7 = 2x - 5 $}$?

Understanding the Concept of { y $}$-intercept

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. In other words, it is the point on the line where the value of x is zero, and the value of y is the { y $}$-intercept. The { y $}$-intercept is an important concept in mathematics, particularly in algebra and geometry.

The Equation of the Line

The given equation of the line is { 4y + 7 = 2x - 5 $}$. To find the { y $}$-intercept, we need to isolate the variable y. We can do this by subtracting 2x from both sides of the equation and then dividing both sides by 4.

Isolating the Variable y

To isolate the variable y, we need to get all the terms involving y on one side of the equation and the constant terms on the other side. We can do this by subtracting 2x from both sides of the equation and then dividing both sides by 4.

{ 4y + 7 = 2x - 5 $}$

Subtracting 2x from both sides:

{ 4y + 7 - 2x = -5 - 2x $}$

Simplifying:

{ 4y - 2x + 7 = -5 - 2x $}$

Adding 2x to both sides:

{ 4y + 7 = -5 $}$

Subtracting 7 from both sides:

{ 4y = -12 $}$

Dividing both sides by 4:

{ y = -3 $}$

Finding the { y $}$-intercept

Now that we have isolated the variable y, we can find the { y $}$-intercept by substituting x = 0 into the equation. Since the { y $}$-intercept is the point on the line where x is equal to zero, we can substitute x = 0 into the equation to find the value of y.

{ y = -3 $}$

Therefore, the { y $}$-intercept of the line { 4y + 7 = 2x - 5 $}$ is -3.

Conclusion

In conclusion, 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. To find the { y $}$-intercept, we need to isolate the variable y by subtracting 2x from both sides of the equation and then dividing both sides by 4. By substituting x = 0 into the equation, we can find the value of y, which is the { y $}$-intercept. In this case, the { y $}$-intercept of the line { 4y + 7 = 2x - 5 $}$ is -3.

Real-World Applications

The concept of { y $}$-intercept has many real-world applications. For example, in economics, the { y $}$-intercept can represent the initial cost of a product or service. In physics, the { y $}$-intercept can represent the initial velocity of an object. In engineering, the { y $}$-intercept can represent the initial displacement of a system.

Common Mistakes

There are several common mistakes that people make when finding the { y $}$-intercept of a line. One common mistake is to forget to isolate the variable y. Another common mistake is to substitute the wrong value of x into the equation. To avoid these mistakes, it is essential to carefully read and understand the equation of the line and to follow the correct steps to isolate the variable y.

Tips and Tricks

Here are some tips and tricks to help you find the { y $}$-intercept of a line:

  • Make sure to isolate the variable y by subtracting 2x from both sides of the equation and then dividing both sides by 4.
  • Substitute x = 0 into the equation to find the value of y.
  • Check your work by plugging the value of y back into the equation to make sure it is true.
  • Use a graphing calculator or a computer program to visualize the line and find the { y $}$-intercept.

Conclusion

In conclusion, the { y $}$-intercept of a line is an important concept in mathematics that has many real-world applications. To find the { y $}$-intercept, we need to isolate the variable y by subtracting 2x from both sides of the equation and then dividing both sides by 4. By substituting x = 0 into the equation, we can find the value of y, which is the { y $}$-intercept. With practice and patience, you can become proficient in finding the { y $}$-intercept of a line.
Q&A: Finding the { y $}$-intercept of a Line

Frequently Asked Questions

Here are some frequently asked questions about finding the { y $}$-intercept of a line:

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: How do I find the { y $}$-intercept of a line?

A: To find the { y $}$-intercept of a line, you need to isolate the variable y by subtracting 2x from both sides of the equation and then dividing both sides by 4. Then, substitute x = 0 into the equation to find the value of y.

Q: What if the equation of the line is not in the form { 4y + 7 = 2x - 5 $}$?

A: If the equation of the line is not in the form { 4y + 7 = 2x - 5 $}$, you need to rewrite the equation in this form before finding the { y $}$-intercept. You can do this by multiplying both sides of the equation by 4 or by dividing both sides of the equation by 2.

Q: Can I use a graphing calculator or a computer program to find the { y $}$-intercept of a line?

A: Yes, you can use a graphing calculator or a computer program to find the { y $}$-intercept of a line. These tools can help you visualize the line and find the { y $}$-intercept quickly and accurately.

Q: What if I make a mistake when finding the { y $}$-intercept of a line?

A: If you make a mistake when finding the { y $}$-intercept of a line, you can try re-reading the equation and re-working the problem. You can also use a graphing calculator or a computer program to check your work and make sure you have the correct answer.

Q: Can I find the { y $}$-intercept of a line if the equation is not linear?

A: No, you cannot find the { y $}$-intercept of a line if the equation is not linear. The { y $}$-intercept is a concept that only applies to linear equations.

Q: What is the significance of the { y $}$-intercept of a line?

A: The { y $}$-intercept of a line is significant because it represents the point at which the line intersects the y-axis. This point is important in many real-world applications, such as economics, physics, and engineering.

Q: Can I use the { y $}$-intercept of a line to make predictions about the behavior of a system?

A: Yes, you can use the { y $}$-intercept of a line to make predictions about the behavior of a system. The { y $}$-intercept can represent the initial condition of a system, and by using the equation of the line, you can make predictions about how the system will behave over time.

Q: What are some common mistakes to avoid when finding the { y $}$-intercept of a line?

A: Some common mistakes to avoid when finding the { y $}$-intercept of a line include:

  • Forgetting to isolate the variable y
  • Substituting the wrong value of x into the equation
  • Not checking your work to make sure the answer is correct
  • Using a graphing calculator or computer program without understanding the underlying math

Conclusion

In conclusion, finding the { y $}$-intercept of a line is an important concept in mathematics that has many real-world applications. By following the steps outlined in this article, you can find the { y $}$-intercept of a line and use it to make predictions about the behavior of a system. Remember to avoid common mistakes and to always check your work to make sure the answer is correct.