Find The Slope Of The Line That Passes Through The Points { (2, -5)$}$ And { (7, 1)$}$.Step 1: Choose { (x_1, Y_1)$} . . . X_1 = 2, Y_1 = -5\$} Step 2 Identify { (x_2, Y_2)$ . \[ .\[ . \[ X_2 = 7, Y_2 =

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Introduction

In mathematics, the slope of a line is a fundamental concept that helps us understand the steepness or incline of a line. It is a crucial concept in geometry, algebra, and calculus. In this article, we will explore how to find the slope of a line that passes through two given points. We will use a step-by-step approach to make it easy to understand and apply.

Step 1: Choose the First Point

To find the slope of a line, we need to choose two points on the line. Let's call these points {(x_1, y_1)$}$ and {(x_2, y_2)$}$. In this example, we will choose the first point as {(2, -5)$}$. This means that {x_1 = 2$}$ and {y_1 = -5$}$.

Step 2: Identify the Second Point

Now that we have chosen the first point, we need to identify the second point. Let's call this point {(x_2, y_2)$}$. In this example, we will choose the second point as {(7, 1)$}$. This means that {x_2 = 7$}$ and {y_2 = 1$}$.

Step 3: Apply the Slope Formula

Now that we have chosen two points, we can apply the slope formula to find the slope of the line. The slope formula is given by:

m=y2βˆ’y1x2βˆ’x1m = \frac{y_2 - y_1}{x_2 - x_1}

where {m$}$ is the slope of the line, and {(x_1, y_1)$}$ and {(x_2, y_2)$}$ are the two points on the line.

Substituting the Values

Now that we have the slope formula, we can substitute the values of {x_1, y_1, x_2,$] and [y_2\$} into the formula. We get:

m=1βˆ’(βˆ’5)7βˆ’2m = \frac{1 - (-5)}{7 - 2}

Simplifying the Expression

Now that we have substituted the values, we can simplify the expression. We get:

m=65m = \frac{6}{5}

Conclusion

In this article, we have learned how to find the slope of a line that passes through two given points. We have used a step-by-step approach to make it easy to understand and apply. We have chosen two points, applied the slope formula, and simplified the expression to find the slope of the line. The slope of the line is {\frac{6}{5}$}$.

Real-World Applications

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

  • Architecture: The slope of a roof is an important consideration in building design.
  • Engineering: The slope of a road or a bridge is critical in ensuring safe and efficient transportation.
  • Surveying: The slope of a land is essential in determining the elevation and orientation of a property.
  • Physics: The slope of a force field is used to describe the direction and magnitude of a force.

Common Mistakes

When finding the slope of a line, there are several common mistakes to avoid:

  • Incorrectly identifying the points: Make sure to choose the correct points on the line.
  • Incorrectly applying the slope formula: Double-check the formula and ensure that you are using the correct values.
  • Not simplifying the expression: Simplify the expression to ensure that you get the correct slope.

Conclusion

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 calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line.

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

A: To find the slope of a line, you need to choose two points on the line, {(x_1, y_1)$}$ and {(x_2, y_2)$}$. Then, you can use the slope formula:

m=y2βˆ’y1x2βˆ’x1m = \frac{y_2 - y_1}{x_2 - x_1}

Q: What is the slope formula?

A: The slope formula is:

m=y2βˆ’y1x2βˆ’x1m = \frac{y_2 - y_1}{x_2 - x_1}

where {m$}$ is the slope of the line, and {(x_1, y_1)$}$ and {(x_2, y_2)$}$ are the two points on the line.

Q: How do I choose the two points on the line?

A: You can choose any two points on the line. However, it's best to choose points that are not too close together, as this will give you a more accurate slope.

Q: What if the two points are the same?

A: If the two points are the same, then the slope of the line is undefined. This is because the denominator of the slope formula would be zero.

Q: What if the two points are not on the same line?

A: If the two points are not on the same line, then the slope of the line is undefined. This is because the two points do not have a common slope.

Q: Can I use the slope formula with negative numbers?

A: Yes, you can use the slope formula with negative numbers. The formula will still work as long as you follow the order of operations (PEMDAS).

Q: Can I use the slope formula with fractions?

A: Yes, you can use the slope formula with fractions. The formula will still work as long as you follow the order of operations (PEMDAS).

Q: How do I simplify the slope formula?

A: To simplify the slope formula, you can cancel out any common factors in the numerator and denominator.

Q: What is the slope of a horizontal line?

A: The slope of a horizontal line is zero. This is because the rise (vertical change) is zero.

Q: What is the slope of a vertical line?

A: The slope of a vertical line is undefined. This is because the run (horizontal change) is zero.

Q: Can I use the slope formula to find the equation of a line?

A: Yes, you can use the slope formula to find the equation of a line. Once you have the slope, you can use the point-slope form of a line to find the equation.

Q: What is the point-slope form of a line?

A: The point-slope form of a line is:

yβˆ’y1=m(xβˆ’x1)y - y_1 = m(x - x_1)

where {m$}$ is the slope of the line, and {(x_1, y_1)$}$ is a point on the line.

Conclusion

In conclusion, finding the slope of a line is a crucial concept in mathematics that has numerous real-world applications. By following the step-by-step approach outlined in this article, you can easily find the slope of a line that passes through two given points. Remember to avoid common mistakes and simplify the expression to ensure that you get the correct slope.