What Is The Slope Of The Line $4x = -3y + 8$?A. 4 B. $-\frac{3}{4}$ C. $-\frac{4}{3}$ D. 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 formula: m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are two points on the line. However, in this article, we will focus on finding the slope of a line given in the form of an equation.

Understanding the Equation of a Line

The equation of a line can be written in the form of y = mx + c, where m is the slope of the line and c is the y-intercept. However, the given equation 4x=−3y+84x = -3y + 8 is not in the standard form of a line. To find the slope, we need to rewrite the equation in the standard form.

Converting the Equation to Standard Form

To convert the equation 4x=−3y+84x = -3y + 8 to the standard form, we need to isolate y on one side of the equation. We can do this by subtracting 8 from both sides of the equation and then dividing both sides by -3.

# Import necessary modules
import sympy as sp

# Define variables
x = sp.symbols('x')
y = sp.symbols('y')

# Define the equation
equation = sp.Eq(4*x, -3*y + 8)

# Solve the equation for y
solution = sp.solve(equation, y)

# Print the solution
print(solution)

Finding the Slope

After solving the equation for y, we get the equation in the form of y = mx + c. We can now identify the slope of the line, which is the coefficient of x in the equation.

# Define the solution
solution = [-4/3*x + 8/3]

# Print the solution
print(solution)

Conclusion

In conclusion, the slope of the line 4x=−3y+84x = -3y + 8 is -4/3. This can be calculated by rewriting the equation in the standard form and identifying the coefficient of x.

Final Answer

The final answer is −43\boxed{-\frac{4}{3}}.

Discussion

The slope of a line is a fundamental concept in mathematics 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 formula: m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are two points on the line. However, in this article, we focused on finding the slope of a line given in the form of an equation.

Step-by-Step Solution

  1. Rewrite the equation 4x=−3y+84x = -3y + 8 in the standard form by isolating y on one side of the equation.
  2. Solve the equation for y to get the equation in the form of y = mx + c.
  3. Identify the slope of the line, which is the coefficient of x in the equation.

Common Mistakes

  • Not rewriting the equation in the standard form before finding the slope.
  • Not identifying the coefficient of x in the equation as the slope.

Real-World Applications

  • Finding the slope of a line is essential in various real-world applications such as physics, engineering, and economics.
  • The slope of a line can be used to model the rate of change of a quantity over time.

Future Work

  • Finding the slope of a line is a fundamental concept in mathematics that has numerous applications in various fields.
  • Further research can be done on finding the slope of a line in different forms of equations.

References

Introduction

In our previous article, we discussed how to find the slope of a line given in the form of an equation. In this article, we will provide a Q&A section to help you better understand the concept of finding the slope of a line.

Q1: What is the slope of a line?

A1: The slope of a line is a measure of how steep the line is and is denoted by the letter 'm'. It represents the rate of change of the line.

Q2: How do I find the slope of a line?

A2: To find the slope of a line, you need to rewrite the equation in the standard form (y = mx + c) and identify the coefficient of x as the slope.

Q3: What is the standard form of a line?

A3: The standard form of a line is y = mx + c, where m is the slope and c is the y-intercept.

Q4: How do I rewrite an equation in the standard form?

A4: To rewrite an equation in the standard form, you need to isolate y on one side of the equation and then identify the coefficient of x as the slope.

Q5: What is the y-intercept?

A5: The y-intercept is the point where the line intersects the y-axis and is denoted by the letter 'c'.

Q6: How do I find the y-intercept?

A6: To find the y-intercept, you need to rewrite the equation in the standard form and identify the constant term as the y-intercept.

Q7: What is the difference between the slope and the y-intercept?

A7: The slope represents the rate of change of the line, while the y-intercept represents the point where the line intersects the y-axis.

Q8: Can I find the slope of a line using the slope-intercept form?

A8: Yes, you can find the slope of a line using the slope-intercept form (y = mx + c) by identifying the coefficient of x as the slope.

Q9: What is the slope-intercept form?

A9: The slope-intercept form is y = mx + c, where m is the slope and c is the y-intercept.

Q10: Can I find the slope of a line using the point-slope form?

A10: Yes, you can find the slope of a line using the point-slope form (y - y1 = m(x - x1)) by identifying the coefficient of x as the slope.

Q11: What is the point-slope form?

A11: The point-slope form is y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line.

Q12: Can I find the slope of a line using the two-point form?

A12: Yes, you can find the slope of a line using the two-point form (y2 - y1 = m(x2 - x1)) by identifying the coefficient of x as the slope.

Q13: What is the two-point form?

A13: The two-point form is y2 - y1 = m(x2 - x1), where (x1, y1) and (x2, y2) are two points on the line.

Q14: Can I find the slope of a line using the equation of a circle?

A14: Yes, you can find the slope of a line using the equation of a circle by rewriting the equation in the standard form and identifying the coefficient of x as the slope.

Q15: What is the equation of a circle?

A15: The equation of a circle is (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center of the circle and r is the radius.

Conclusion

In conclusion, finding the slope of a line is a fundamental concept in mathematics that has numerous applications in various fields. By understanding the concept of slope and how to find it, you can solve a wide range of problems in mathematics and other subjects.

Final Answer

The final answer is −43\boxed{-\frac{4}{3}}.

Discussion

The slope of a line is a measure of how steep the line is and is denoted by the letter 'm'. It represents the rate of change of the line. The slope of a line can be found using various forms of equations, including the standard form, slope-intercept form, point-slope form, and two-point form.

Step-by-Step Solution

  1. Rewrite the equation in the standard form (y = mx + c).
  2. Identify the coefficient of x as the slope.
  3. Use the slope-intercept form (y = mx + c) to find the slope.
  4. Use the point-slope form (y - y1 = m(x - x1)) to find the slope.
  5. Use the two-point form (y2 - y1 = m(x2 - x1)) to find the slope.

Common Mistakes

  • Not rewriting the equation in the standard form before finding the slope.
  • Not identifying the coefficient of x as the slope.
  • Not using the correct form of the equation to find the slope.

Real-World Applications

  • Finding the slope of a line is essential in various real-world applications such as physics, engineering, and economics.
  • The slope of a line can be used to model the rate of change of a quantity over time.

Future Work

  • Finding the slope of a line is a fundamental concept in mathematics that has numerous applications in various fields.
  • Further research can be done on finding the slope of a line in different forms of equations.

References