Determine The Intercepts Of The Line. Do Not Round Your Answers.$ \begin{array}{l} y - 4 = 7(x - 6) \ x \text{-intercept (\square, \square) \ y \text -intercept (\square, \square) \end{array} }$

Understanding the Problem

To determine the intercepts of a line, we need to find the points where the line intersects the x-axis and the y-axis. The x-intercept is the point where the line crosses the x-axis, and the y-intercept is the point where the line crosses the y-axis.

The Given Equation

The given equation is:

${ y - 4 = 7(x - 6) }$

This is a linear equation in the slope-intercept form, where the slope is 7 and the y-intercept is 4.

Finding the X-Intercept

To find the x-intercept, we need to set y = 0 and solve for x.

${ 0 - 4 = 7(x - 6) }$

${ -4 = 7(x - 6) }$

${ -4 = 7x - 42 }$

${ 7x = 38 }$

${ x = \frac{38}{7} }$

So, the x-intercept is:

${ (\frac{38}{7}, 0) }$

Finding the Y-Intercept

To find the y-intercept, we need to set x = 0 and solve for y.

${ y - 4 = 7(0 - 6) }$

${ y - 4 = -42 }$

${ y = -42 + 4 }$

${ y = -38 }$

So, the y-intercept is:

${ (0, -38) }$

Conclusion

In this problem, we determined the x-intercept and the y-intercept of the line using the given equation. The x-intercept is the point where the line crosses the x-axis, and the y-intercept is the point where the line crosses the y-axis. We found that the x-intercept is (\frac{38}{7}, 0) and the y-intercept is (0, -38).

Key Takeaways

• To find the x-intercept, set y = 0 and solve for x.
• To find the y-intercept, set x = 0 and solve for y.
• The x-intercept is the point where the line crosses the x-axis, and the y-intercept is the point where the line crosses the y-axis.

Real-World Applications

Understanding the intercepts of a line is crucial in various real-world applications, such as:

• Physics: To determine the position and velocity of an object at a given time.
• Engineering: To design and optimize systems, such as electrical circuits and mechanical systems.
• Economics: To analyze and predict economic trends and patterns.

Common Mistakes

• Rounding answers: Make sure to provide exact answers, without rounding.
• Incorrectly solving equations: Double-check your work and ensure that you are solving the equations correctly.

Tips and Tricks

• Use algebraic manipulations: Use algebraic manipulations to simplify and solve equations.
• Visualize the problem: Visualize the problem and the line to better understand the intercepts.
• Check your work: Double-check your work and ensure that you are providing accurate answers.
  Determine the Intercepts of the Line: Q&A =====================================

Q: What are the intercepts of a line?

A: The intercepts of a line are the points where the line intersects the x-axis and the y-axis. The x-intercept is the point where the line crosses the x-axis, and the y-intercept is the point where the line crosses the y-axis.

Q: How do I find the x-intercept of a line?

A: To find the x-intercept, set y = 0 and solve for x. This will give you the point where the line crosses the x-axis.

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

A: To find the y-intercept, set x = 0 and solve for y. This will give you the point where the line crosses the y-axis.

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

A: The equation of a line in slope-intercept form is:

${ y = mx + b }$

where m is the slope and b is the y-intercept.

Q: How do I determine the slope of a line?

A: To determine the slope of a line, you can use the formula:

${ m = \frac{y_2 - y_1}{x_2 - x_1} }$

where (x1, y1) and (x2, y2) are two points on the line.

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

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

Q: Can I have a linear equation with no x-intercept?

A: Yes, it is possible to have a linear equation with no x-intercept. This occurs when the slope of the line is zero, and the y-intercept is not zero.

Q: Can I have a linear equation with no y-intercept?

A: Yes, it is possible to have a linear equation with no y-intercept. This occurs when the slope of the line is not zero, and the y-intercept is zero.

Q: How do I graph a linear equation?

A: To graph a linear equation, you can use the slope-intercept form and plot the y-intercept. Then, use the slope to determine the direction and steepness of the line.

Q: What are some common mistakes to avoid when finding intercepts?

A: Some common mistakes to avoid when finding intercepts include:

• Rounding answers
• Incorrectly solving equations
• Not checking work

Q: What are some real-world applications of finding intercepts?

A: Some real-world applications of finding intercepts include:

• Physics: To determine the position and velocity of an object at a given time.
• Engineering: To design and optimize systems, such as electrical circuits and mechanical systems.
• Economics: To analyze and predict economic trends and patterns.

Q: How do I determine the equation of a line given two points?

A: To determine the equation of a line given two points, you can use the slope formula and the point-slope form of a linear equation.

Q: Can I have a linear equation with multiple x-intercepts?

A: No, a linear equation can only have one x-intercept. However, it can have multiple y-intercepts.

Q: Can I have a linear equation with multiple y-intercepts?

A: Yes, a linear equation can have multiple y-intercepts. This occurs when the slope of the line is zero, and the y-intercept is not zero.