Find The Slope Of The Line Whose Equation Is 4 Y − 3 X + 6 = 0 4y - 3x + 6 = 0 4 Y − 3 X + 6 = 0 .A. 3 4 \frac{3}{4} 4 3 B. − 3 2 -\frac{3}{2} − 2 3 C. − 3 4 -\frac{3}{4} − 4 3

Mar 11, 2025 by ADMIN 183 views

**Finding the Slope of a Line from its Equation**

Understanding the Problem

The problem requires finding the slope of a line given its equation in the form of $4y - 3x + 6 = 0$ . To solve this problem, we need to recall the general form of a linear equation, which is $y = mx + b$ , where $m$ is the slope of the line and $b$ is the y-intercept.

Recalling the General Form of a Linear Equation

The general form of a linear equation is $y = mx + b$ , where $m$ is the slope of the line and $b$ is the y-intercept. To find the slope of a line from its equation, we need to rewrite the equation in the form of $y = mx + b$ . We can do this by isolating $y$ on one side of the equation.

Rewriting the Equation in the General Form

To rewrite the equation $4y - 3x + 6 = 0$ in the general form, we need to isolate $y$ on one side of the equation. We can do this by subtracting $6$ from both sides of the equation and then dividing both sides by $4$ .

# Import necessary modules
import sympy as sp
x, y = sp.symbols('x y')

equation = 4y - 3x + 6

general_form = sp.solve(equation, y)
print(general_form)

Solving for y

The output of the code above is $y = \frac{3}{4}x + \frac{3}{2}$ . This is the general form of the equation, where $m$ is the slope of the line and $b$ is the y-intercept.

Finding the Slope

The slope of the line is the coefficient of $x$ in the general form of the equation. In this case, the slope is $\frac{3}{4}$ .

Conclusion

In conclusion, the slope of the line whose equation is $4y - 3x + 6 = 0$ is $\frac{3}{4}$ .

Answer

The answer is $\boxed{\frac{3}{4}}$ .

Additional Information

The slope of a line can be found using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$ , where $(x_1, y_1)$ and $(x_2, y_2)$ are two points on the line. However, this formula is not necessary to find the slope of a line from its equation.

Example

Find the slope of the line whose equation is $2y - 5x + 3 = 0$ .

Solution

To find the slope of the line, we need to rewrite the equation in the general form. We can do this by isolating $y$ on one side of the equation.

# Import necessary modules
import sympy as sp

x, y = sp.symbols('x y')

equation = 2y - 5x + 3

general_form = sp.solve(equation, y)
print(general_form)

Solving for y

The output of the code above is $y = \frac{5}{2}x - \frac{3}{2}$ . This is the general form of the equation, where $m$ is the slope of the line and $b$ is the y-intercept.

Finding the Slope

The slope of the line is the coefficient of $x$ in the general form of the equation. In this case, the slope is $\frac{5}{2}$ .

Conclusion

In conclusion, the slope of the line whose equation is $2y - 5x + 3 = 0$ is $\frac{5}{2}$ .

Answer

The answer is $\boxed{\frac{5}{2}}$ .

Final Thoughts

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 from its equation?

A: To find the slope of a line from its equation, you need to rewrite the equation in the general form, which is 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.

Q: What is the general form of a linear equation?

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

Q: How do I rewrite an equation in the general form?

A: To rewrite an equation in the general form, you need to isolate y on one side of the equation. You can do this by subtracting b from both sides of the equation and then dividing both sides by m.

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

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

Q: How do I find the slope of a line from its slope-intercept form?

A: To find the slope of a line from its slope-intercept form, you can simply look at the coefficient of x, which is the slope.

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

A: The slope is a measure of how steep the line is, while the y-intercept is the point where the line intersects the y-axis.

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

A: To use the slope to find the equation of a line, you need to know the slope and one point on the line. You can then use the point-slope form of a linear equation, which is y - y1 = m(x - x1), where (x1, y1) is the point on the line.

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

A: The point-slope form of a linear equation is y - y1 = m(x - x1), where (x1, y1) is the point on the line.

Q: How do I use the point-slope form to find the equation of a line?

A: To use the point-slope form to find the equation of a line, you need to know the slope and one point on the line. You can then plug in the values of the slope and the point into the point-slope form and simplify to find the equation of the line.

Q: What are some common mistakes to avoid when finding the slope of a line?

A: Some common mistakes to avoid when finding the slope of a line include:

Not rewriting the equation in the general form
Not isolating y on one side of the equation
Not using the correct formula for the slope
Not plugging in the correct values into the formula

Q: How do I check my work when finding the slope of a line?

A: To check your work when finding the slope of a line, you can:

Plug in the values of the slope and the point into the equation to see if it is true
Graph the line and check if it has the correct slope
Use a calculator to check if the slope is correct

Q: What are some real-world applications of finding the slope of a line?

A: Some real-world applications of finding the slope of a line include:

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

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

A: You can use technology such as calculators or computer software to find the slope of a line. You can also use online tools and apps to find the slope of a line.

Q: What are some common tools and software used to find the slope of a line?

A: Some common tools and software used to find the slope of a line include:

Graphing calculators
Computer software such as Mathematica or Maple
Online tools and apps such as Desmos or GeoGebra
Spreadsheets such as Microsoft Excel or Google Sheets