Understanding the Basics of Slope and Line Equations

Oct 13, 2025 by vinooalluri.com 52 views

What is the Slope of the Line Whose Equation is $5y + 6x - 2 = 0$ ?

Understanding the Basics of Slope and Line Equations

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 measure of how much the line rises (or falls) vertically over a given horizontal distance. The slope of a line is denoted by the letter 'm' and is calculated using the formula: m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are two points on the line.

The Equation of a Line

The equation of a line is a mathematical expression that describes the relationship between the x and y coordinates of points on the line. It is typically written in the form: Ax + By + C = 0, where A, B, and C are constants. In the given equation, $5y + 6x - 2 = 0$ , we can see that A = 6, B = -5, and C = -2.

Slope-Intercept Form

One of the most common forms of a line equation is the slope-intercept form, which is written as: y = mx + b, where m is the slope and b is the y-intercept. To find the slope of the line, we need to rewrite the given equation in slope-intercept form.

Rewriting the Equation in Slope-Intercept Form

To rewrite the equation in slope-intercept form, we need to isolate y on one side of the equation. We can do this by subtracting 6x from both sides of the equation and then dividing both sides by -5.

$5y + 6x - 2 = 0$

Subtracting 6x from both sides:

$5y - 2 = -6x$

Adding 2 to both sides:

$5y = -6x + 2$

Dividing both sides by -5:

$y = \frac{-6x + 2}{5}$

$y = \frac{-6}{5}x + \frac{2}{5}$

Finding the Slope

Now that we have rewritten the equation in slope-intercept form, we can easily identify the slope. The slope is the coefficient of x, which is -6/5.

Conclusion

In conclusion, the slope of the line whose equation is $5y + 6x - 2 = 0$ is -6/5. This means that for every unit increase in x, y decreases by 6/5 units.

Answer

The correct answer is C. Slope = -6/5.

Additional Tips and Tricks

When rewriting the equation in slope-intercept form, make sure to isolate y on one side of the equation.
The slope is the coefficient of x in the slope-intercept form of the equation.
The y-intercept is the constant term in the slope-intercept form of the equation.

Real-World Applications

Understanding the slope of a line has many real-world applications, such as:

Calculating the steepness of a roof or a hill
Determining the rate of change of a quantity over time
Finding the equation of a line that passes through two points

Common Mistakes

Failing to isolate y on one side of the equation when rewriting the equation in slope-intercept form
Misidentifying the slope or y-intercept in the slope-intercept form of the equation
Not checking the units of the slope and y-intercept

Conclusion

In conclusion, finding the slope of a line is a crucial concept in mathematics that has many real-world applications. By understanding the basics of slope and line equations, we can easily identify the slope of a line and apply it to various problems.
Frequently Asked Questions (FAQs) About Slope and Line Equations

Q: What is the slope of a line?

A: The slope of a line is a measure of how much the line rises (or falls) vertically over a given horizontal distance. It is denoted by the letter 'm' and is calculated using the formula: m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are 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 rewrite the equation of the line in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept. You can do this by isolating y on one side of the equation and then identifying the coefficient of x as the slope.

Q: What is the difference between slope and y-intercept?

A: The slope (m) is the coefficient of x in the slope-intercept form of the equation, while the y-intercept (b) is the constant term in the slope-intercept form of the equation. The slope tells you how much the line rises (or falls) vertically over a given horizontal distance, while the y-intercept tells you where the line intersects the y-axis.

Q: How do I determine the equation of a line that passes through two points?

A: To determine the equation of a line that passes through two points, you need to find the slope of the line using the formula: m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the two points. Then, you can use the point-slope form of the equation (y - y1 = m(x - x1)) to find the equation of the line.

Q: What is the point-slope form of the equation?

A: The point-slope form of the equation is y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope. This form is useful for finding the equation of a line that passes through two points.

Q: How do I convert the point-slope form to the slope-intercept form?

A: To convert the point-slope form to the slope-intercept form, you need to isolate y on one side of the equation. You can do this by adding y1 to both sides of the equation and then subtracting m(x - x1) from both sides.

Q: What is the equation of a line that passes through the points (2, 3) and (4, 5)?

A: To find the equation of a line that passes through the points (2, 3) and (4, 5), you need to find the slope of the line using the formula: m = (y2 - y1) / (x2 - x1). Then, you can use the point-slope form of the equation to find the equation of the line.

m = (5 - 3) / (4 - 2) m = 2 / 2 m = 1

y - 3 = 1(x - 2)

y - 3 = x - 2

y = x - 2 + 3

y = x + 1

Q: What is the equation of a line that passes through the points (0, 2) and (3, 4)?

A: To find the equation of a line that passes through the points (0, 2) and (3, 4), you need to find the slope of the line using the formula: m = (y2 - y1) / (x2 - x1). Then, you can use the point-slope form of the equation to find the equation of the line.

m = (4 - 2) / (3 - 0) m = 2 / 3

y - 2 = 2/3(x - 0)

y - 2 = 2/3x

y = 2/3x + 2

Conclusion