What Is The Slope Of The Line Whose Equation Is Y − 4 = 5 2 ( X − 2 Y - 4 = \frac{5}{2}(x - 2 Y − 4 = 2 5 ​ ( X − 2 ]?

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Introduction

In mathematics, the slope of a line is a fundamental concept that represents the rate of change of the line. It is a measure of how steep the line is and is denoted by the letter 'm'. The slope of a line can be calculated using the slope-intercept form of a linear equation, which is given by y = mx + b, where 'm' is the slope and 'b' is the y-intercept. In this article, we will discuss how to find the slope of a line whose equation is given in a different form.

Understanding the Equation

The given equation is y−4=52(x−2)y - 4 = \frac{5}{2}(x - 2). This equation is not in the slope-intercept form, but we can rewrite it in a way that will allow us to find the slope. To do this, we need to isolate the variable 'y' on one side of the equation.

Isolating the Variable 'y'

To isolate the variable 'y', we need to get rid of the constant term on the left-hand side of the equation. We can do this by adding 4 to both sides of the equation. This gives us:

y−4+4=52(x−2)y - 4 + 4 = \frac{5}{2}(x - 2)

Simplifying the left-hand side of the equation, we get:

y=52(x−2)y = \frac{5}{2}(x - 2)

Finding the Slope

Now that we have the equation in a form that allows us to find the slope, we can see that the slope is the coefficient of the variable 'x'. In this case, the coefficient of 'x' is 52\frac{5}{2}, which is the slope of the line.

Conclusion

In conclusion, the slope of the line whose equation is y−4=52(x−2)y - 4 = \frac{5}{2}(x - 2) is 52\frac{5}{2}. This is a fundamental concept in mathematics and is used to describe the rate of change of a line.

Slope-Intercept Form

The slope-intercept form of a linear equation is given by y = mx + b, where 'm' is the slope and 'b' is the y-intercept. This form is useful for finding the slope of a line, as the slope is the coefficient of the variable 'x'.

Finding the Y-Intercept

To find the y-intercept of the line, we need to substitute x = 0 into the equation. This gives us:

y=52(0−2)y = \frac{5}{2}(0 - 2)

Simplifying the equation, we get:

y=−5y = -5

Conclusion

In conclusion, the y-intercept of the line is -5. This is the point at which the line intersects the y-axis.

Graphing the Line

To graph the line, we need to plot two points on the coordinate plane. We can do this by substituting x = 0 and x = 1 into the equation. This gives us:

y=52(0−2)=−5y = \frac{5}{2}(0 - 2) = -5

y=52(1−2)=−52y = \frac{5}{2}(1 - 2) = -\frac{5}{2}

Plotting these two points on the coordinate plane, we get a line with a slope of 52\frac{5}{2} and a y-intercept of -5.

Conclusion

In conclusion, the line whose equation is y−4=52(x−2)y - 4 = \frac{5}{2}(x - 2) has a slope of 52\frac{5}{2} and a y-intercept of -5. This is a fundamental concept in mathematics and is used to describe the rate of change of a line.

Real-World Applications

The concept of slope is used in many real-world applications, such as:

  • Physics: The slope of a line can be used to describe the rate of change of an object's velocity.
  • Engineering: The slope of a line can be used to describe the rate of change of a system's output.
  • Economics: The slope of a line can be used to describe the rate of change of a country's GDP.

Conclusion

In conclusion, the slope of a line is a fundamental concept in mathematics that is used to describe the rate of change of a line. It is a measure of how steep the line is and is denoted by the letter 'm'. The slope of a line can be calculated using the slope-intercept form of a linear equation, which is given by y = mx + b, where 'm' is the slope and 'b' is the y-intercept. In this article, we discussed how to find the slope of a line whose equation is given in a different form.

Final Thoughts

The concept of slope is a fundamental concept in mathematics that is used to describe the rate of change of a line. It is a measure of how steep the line is and is denoted by the letter 'm'. The slope of a line can be calculated using the slope-intercept form of a linear equation, which is given by y = mx + b, where 'm' is the slope and 'b' is the y-intercept. In this article, we discussed how to find the slope of a line whose equation is given in a different form. We also discussed the real-world applications of the concept of slope and how it is used in physics, engineering, and economics.

Q: What is the slope of a line?

A: The slope of a line is a measure of how steep the line is. It is denoted by the letter 'm' and is calculated using the slope-intercept form of a linear equation, which is given by y = mx + b, where 'm' is the slope and 'b' is the y-intercept.

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

A: To find the slope of a line, you need to rewrite the equation in the slope-intercept form, which is given by y = mx + b. The slope is then the coefficient of the variable 'x'.

Q: What is the difference between slope and rate of change?

A: The slope and rate of change are related but distinct concepts. The slope is a measure of how steep a line is, while the rate of change is a measure of how quickly a quantity changes.

Q: Can a line have a negative slope?

A: Yes, a line can have a negative slope. A negative slope indicates that the line is decreasing as the value of 'x' increases.

Q: Can a line have a zero slope?

A: Yes, a line can have a zero slope. A zero slope indicates that the line is horizontal and does not change as the value of 'x' increases.

Q: Can a line have an undefined slope?

A: Yes, a line can have an undefined slope. An undefined slope indicates that the line is vertical and does not change as the value of 'y' increases.

Q: How do I graph a line with a given slope?

A: To graph a line with a given slope, you need to use the slope-intercept form of a linear equation, which is given by y = mx + b. You can then plot two points on the coordinate plane and draw a line through them.

Q: How do I find the equation of a line with a given slope?

A: To find the equation of a line with a given slope, you need to use the point-slope form of a linear equation, which is given by y - y1 = m(x - x1), where (x1, y1) is a point on the line and 'm' is the slope.

Q: Can a line have multiple slopes?

A: No, a line can only have one slope. The slope is a measure of how steep the line is, and it is determined by the equation of the line.

Q: How do I determine if a line is increasing or decreasing?

A: To determine if a line is increasing or decreasing, you need to look at the slope of the line. If the slope is positive, the line is increasing. If the slope is negative, the line is decreasing.

Q: Can a line have a slope of 1?

A: Yes, a line can have a slope of 1. A slope of 1 indicates that the line is increasing at a rate of 1 unit for every 1 unit increase in 'x'.

Q: Can a line have a slope of -1?

A: Yes, a line can have a slope of -1. A slope of -1 indicates that the line is decreasing at a rate of 1 unit for every 1 unit increase in 'x'.

Q: How do I find the equation of a line with a given slope and a given point?

A: To find the equation of a line with a given slope and a given point, you need to use the point-slope form of a linear equation, which is given by y - y1 = m(x - x1), where (x1, y1) is the given point and 'm' is the given slope.

Q: Can a line have a slope of 0 and still be a line?

A: Yes, a line can have a slope of 0 and still be a line. A slope of 0 indicates that the line is horizontal and does not change as the value of 'x' increases.

Q: Can a line have a slope of infinity and still be a line?

A: No, a line cannot have a slope of infinity. A slope of infinity indicates that the line is vertical and does not change as the value of 'y' increases.

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

A: To find the equation of a line with a given slope and a given y-intercept, you need to use the slope-intercept form of a linear equation, which is given by y = mx + b, where 'm' is the given slope and 'b' is the given y-intercept.

Q: Can a line have a slope of 1 and a y-intercept of 0?

A: Yes, a line can have a slope of 1 and a y-intercept of 0. This is the equation of the line y = x.

Q: Can a line have a slope of -1 and a y-intercept of 0?

A: Yes, a line can have a slope of -1 and a y-intercept of 0. This is the equation of the line y = -x.

Q: How do I find the equation of a line with a given slope and a given x-intercept?

A: To find the equation of a line with a given slope and a given x-intercept, you need to use the slope-intercept form of a linear equation, which is given by y = mx + b, where 'm' is the given slope and the x-intercept is given by the equation x = -b/m.

Q: Can a line have a slope of 1 and an x-intercept of 1?

A: Yes, a line can have a slope of 1 and an x-intercept of 1. This is the equation of the line y = x - 1.

Q: Can a line have a slope of -1 and an x-intercept of 1?

A: Yes, a line can have a slope of -1 and an x-intercept of 1. This is the equation of the line y = -x + 1.

Q: How do I find the equation of a line with a given slope and a given point and a given y-intercept?

A: To find the equation of a line with a given slope and a given point and a given y-intercept, you need to use the point-slope form of a linear equation, which is given by y - y1 = m(x - x1), where (x1, y1) is the given point and 'm' is the given slope. You can then substitute the given y-intercept into the equation to find the equation of the line.

Q: Can a line have a slope of 1, a given point, and a given y-intercept?

A: Yes, a line can have a slope of 1, a given point, and a given y-intercept. This is the equation of the line y = x + b, where (x, y) is the given point and 'b' is the given y-intercept.

Q: Can a line have a slope of -1, a given point, and a given y-intercept?

A: Yes, a line can have a slope of -1, a given point, and a given y-intercept. This is the equation of the line y = -x + b, where (x, y) is the given point and 'b' is the given y-intercept.

Q: How do I find the equation of a line with a given slope and a given x-intercept and a given y-intercept?

A: To find the equation of a line with a given slope and a given x-intercept and a given y-intercept, you need to use the slope-intercept form of a linear equation, which is given by y = mx + b, where 'm' is the given slope and the x-intercept and y-intercept are given by the equations x = -b/m and y = b.

Q: Can a line have a slope of 1, a given x-intercept, and a given y-intercept?

A: Yes, a line can have a slope of 1, a given x-intercept, and a given y-intercept. This is the equation of the line y = x + b, where x = -b/m and y = b.

Q: Can a line have a slope of -1, a given x-intercept, and a given y-intercept?

A: Yes, a line can have a slope of -1, a given x-intercept, and a given y-intercept. This is the equation of the line y = -x + b, where x = -b/m and y = b.

Q: How do I find the equation of a line with a given slope and a given point and a given x-intercept?

A: To find the equation of a line with a given slope and a given point and a given x-intercept, you need to use the point-slope form of a linear equation, which is given by y - y1 = m(x - x1), where (x1, y1) is the given point and 'm' is the given slope. You can then substitute the given x-intercept into the equation to find the equation of the line.

Q: Can a line have a slope of 1, a given point, and a given x-intercept?

A: Yes, a line can have a slope of 1, a given point, and a given x-intercept. This is the equation of the line y = x + b, where (x, y) is the given