Find A Point On The Line And The Line's Slope.Given Equation: Y + 2 = 1 4 ( X + 5 Y + 2 = \frac{1}{4}(x + 5 Y + 2 = 4 1 ​ ( X + 5 ]Point On The Line: ( ⟦ , □ (\llbracket, \square ( [ [ , □ ]Slope: □ \square □

by ADMIN 209 views

=====================================================

Understanding the Problem

To find a point on the line and the line's slope, we need to analyze the given equation and identify the necessary information. The equation provided is in the form of y+2=14(x+5)y + 2 = \frac{1}{4}(x + 5). Our goal is to determine a point on the line and the slope of the line.

Identifying the Point on the Line

To find a point on the line, we need to identify the coordinates of the point. The coordinates of a point on the line can be found by substituting a value of xx into the equation and solving for yy. Let's choose a value of xx and substitute it into the equation.

Choosing a Value of xx

For simplicity, let's choose x=0x = 0. Substituting x=0x = 0 into the equation, we get:

y+2=14(0+5)y + 2 = \frac{1}{4}(0 + 5)

Simplifying the equation, we get:

y+2=54y + 2 = \frac{5}{4}

Subtracting 2 from both sides, we get:

y=54−2y = \frac{5}{4} - 2

Simplifying further, we get:

y=54−84y = \frac{5}{4} - \frac{8}{4}

Combining the fractions, we get:

y=−34y = -\frac{3}{4}

So, the point on the line is (0,−34)(0, -\frac{3}{4}).

Identifying the Slope of the Line

To find the slope of the line, we need to identify the coefficient of xx in the equation. The coefficient of xx is 14\frac{1}{4}, which represents the slope of the line.

Understanding the Slope

The slope of a line is a measure of how steep the line is. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run). In this case, the slope is 14\frac{1}{4}, which means that for every 1 unit of horizontal change, the line rises by 14\frac{1}{4} unit.

Conclusion

In conclusion, we have found a point on the line and the slope of the line. The point on the line is (0,−34)(0, -\frac{3}{4}), and the slope of the line is 14\frac{1}{4}. This information can be used to graph the line and understand its behavior.

Graphing the Line

To graph the line, we can use the point-slope form of a linear equation, which is y−y1=m(x−x1)y - y_1 = m(x - x_1), where (x1,y1)(x_1, y_1) is a point on the line and mm is the slope. Substituting the values we found earlier, we get:

y−(−34)=14(x−0)y - (-\frac{3}{4}) = \frac{1}{4}(x - 0)

Simplifying the equation, we get:

y+34=14xy + \frac{3}{4} = \frac{1}{4}x

Subtracting 34\frac{3}{4} from both sides, we get:

y=14x−34y = \frac{1}{4}x - \frac{3}{4}

This is the equation of the line in slope-intercept form, which is y=mx+by = mx + b, where mm is the slope and bb is the y-intercept.

Graphing the Line

To graph the line, we can plot the point (0,−34)(0, -\frac{3}{4}) and use the slope to determine the direction of the line. Since the slope is 14\frac{1}{4}, the line will rise by 14\frac{1}{4} unit for every 1 unit of horizontal change.

Plotting the Point

To plot the point (0,−34)(0, -\frac{3}{4}), we can start at the origin (0, 0) and move 34\frac{3}{4} units down. This will give us the point (0,−34)(0, -\frac{3}{4}).

Determining the Direction of the Line

To determine the direction of the line, we can use the slope to determine the direction of the line. Since the slope is 14\frac{1}{4}, the line will rise by 14\frac{1}{4} unit for every 1 unit of horizontal change. This means that the line will have a gentle slope and will rise slowly as we move to the right.

Conclusion

In conclusion, we have graphed the line and understood its behavior. The line has a point on it at (0,−34)(0, -\frac{3}{4}) and a slope of 14\frac{1}{4}. This information can be used to understand the behavior of the line and to make predictions about its behavior.

Real-World Applications

The concept of finding a point on a line and the line's slope has many real-world applications. For example, in physics, the slope of a line can be used to determine the velocity of an object. In engineering, the slope of a line can be used to determine the angle of a ramp or a slope.

Example 1: Velocity of an Object

Suppose we have an object moving at a constant velocity. We can use the slope of the line to determine the velocity of the object. If the slope of the line is 14\frac{1}{4}, then the velocity of the object is 14\frac{1}{4} unit per unit of time.

Example 2: Angle of a Ramp

Suppose we have a ramp with a slope of 14\frac{1}{4}. We can use the slope of the line to determine the angle of the ramp. If the slope of the line is 14\frac{1}{4}, then the angle of the ramp is tan⁡−1(14)\tan^{-1}(\frac{1}{4}).

Conclusion

In conclusion, we have discussed the concept of finding a point on a line and the line's slope. We have used the equation y+2=14(x+5)y + 2 = \frac{1}{4}(x + 5) to find a point on the line and the slope of the line. We have also graphed the line and understood its behavior. The concept of finding a point on a line and the line's slope has many real-world applications, including determining the velocity of an object and the angle of a ramp.

=====================================================================================

Q: What is the equation of the line?

A: The equation of the line is y+2=14(x+5)y + 2 = \frac{1}{4}(x + 5).

Q: How do I find a point on the line?

A: To find a point on the line, you need to substitute a value of xx into the equation and solve for yy. Let's choose x=0x = 0 and substitute it into the equation. We get:

y+2=14(0+5)y + 2 = \frac{1}{4}(0 + 5)

Simplifying the equation, we get:

y+2=54y + 2 = \frac{5}{4}

Subtracting 2 from both sides, we get:

y=54−2y = \frac{5}{4} - 2

Simplifying further, we get:

y=54−84y = \frac{5}{4} - \frac{8}{4}

Combining the fractions, we get:

y=−34y = -\frac{3}{4}

So, the point on the line is (0,−34)(0, -\frac{3}{4}).

Q: How do I find the slope of the line?

A: To find the slope of the line, you need to identify the coefficient of xx in the equation. The coefficient of xx is 14\frac{1}{4}, which represents the slope of the line.

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

A: The slope-intercept form of a linear equation is y=mx+by = mx + b, where mm is the slope and bb is the y-intercept.

Q: How do I graph the line?

A: To graph the line, you can use the point-slope form of a linear equation, which is y−y1=m(x−x1)y - y_1 = m(x - x_1), where (x1,y1)(x_1, y_1) is a point on the line and mm is the slope. Substituting the values we found earlier, we get:

y−(−34)=14(x−0)y - (-\frac{3}{4}) = \frac{1}{4}(x - 0)

Simplifying the equation, we get:

y+34=14xy + \frac{3}{4} = \frac{1}{4}x

Subtracting 34\frac{3}{4} from both sides, we get:

y=14x−34y = \frac{1}{4}x - \frac{3}{4}

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

Q: What are some real-world applications of finding a point on the line and the line's slope?

A: The concept of finding a point on the line and the line's slope has many real-world applications, including determining the velocity of an object and the angle of a ramp.

Q: How do I determine the velocity of an object?

A: To determine the velocity of an object, you can use the slope of the line. If the slope of the line is 14\frac{1}{4}, then the velocity of the object is 14\frac{1}{4} unit per unit of time.

Q: How do I determine the angle of a ramp?

A: To determine the angle of a ramp, you can use the slope of the line. If the slope of the line is 14\frac{1}{4}, then the angle of the ramp is tan⁡−1(14)\tan^{-1}(\frac{1}{4}).

Q: What are some common mistakes to avoid when finding a point on the line and the line's slope?

A: Some common mistakes to avoid when finding a point on the line and the line's slope include:

  • Not substituting a value of xx into the equation to find a point on the line.
  • Not identifying the coefficient of xx in the equation to find the slope of the line.
  • Not using the point-slope form of a linear equation to graph the line.
  • Not considering the real-world applications of finding a point on the line and the line's slope.

Q: How can I practice finding a point on the line and the line's slope?

A: You can practice finding a point on the line and the line's slope by:

  • Substituting different values of xx into the equation to find different points on the line.
  • Identifying the coefficient of xx in the equation to find the slope of the line.
  • Graphing the line using the point-slope form of a linear equation.
  • Considering the real-world applications of finding a point on the line and the line's slope.

Q: What are some resources for learning more about finding a point on the line and the line's slope?

A: Some resources for learning more about finding a point on the line and the line's slope include:

  • Online tutorials and videos
  • Math textbooks and workbooks
  • Online math communities and forums
  • Math apps and software

Conclusion

In conclusion, finding a point on the line and the line's slope is an important concept in mathematics that has many real-world applications. By understanding the equation of the line, finding a point on the line, and identifying the slope of the line, you can graph the line and understand its behavior. Additionally, by considering the real-world applications of finding a point on the line and the line's slope, you can develop a deeper understanding of the concept and its importance in various fields.