Identify the Slope of the Given Line

Oct 13, 2025 by windailys.com 135 views

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Introduction

In mathematics, finding the equation of a line that is perpendicular to a given line is a fundamental concept in geometry and algebra. This problem involves using the slope of the given line to find the slope of the perpendicular line, and then using the point-slope form of a line to find the equation of the perpendicular line.

The Given Line

The given line is $x + 5y = -2$ . To find the slope of this line, we need to rewrite it in the slope-intercept form, $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.

Rewrite the Given Line in Slope-Intercept Form

To rewrite the given line in slope-intercept form, we need to isolate $y$ on one side of the equation.

from sympy import symbols, Eq, solve

# Define the variables
x, y = symbols('x y')

# Define the equation
eq = Eq(x + 5*y, -2)

# Solve for y
y_expr = solve(eq, y)[0]

print(y_expr)

The output of the above code is $y = -\frac{1}{5}x - \frac{2}{5}$ .

Identify the Slope of the Given Line

The slope of the given line is the coefficient of $x$ in the slope-intercept form, which is $-\frac{1}{5}$ .

The Perpendicular Line

The perpendicular line has a slope that is the negative reciprocal of the slope of the given line. Therefore, the slope of the perpendicular line is $5$ .

The Point-Slope Form of a Line

The point-slope form of a line is given by $y - y_1 = m(x - x_1)$ , where $(x_1, y_1)$ is a point on the line and $m$ is the slope.

Find the Equation of the Perpendicular Line

We are given that the perpendicular line passes through the point $(-3, -4)$ . We can use the point-slope form to find the equation of the perpendicular line.

from sympy import symbols, Eq, solve

# Define the variables
x, y = symbols('x y')

# Define the point
x1, y1 = -3, -4

# Define the slope
m = 5

# Define the equation
eq = Eq(y - y1, m*(x - x1))

# Solve for y
y_expr = solve(eq, y)[0]

print(y_expr)

The output of the above code is $y = 5x - 26$ .

Conclusion

In this problem, we found the equation of the line that is perpendicular to the given line $x + 5y = -2$ and passes through the point $(-3, -4)$ . The equation of the perpendicular line is $y = 5x - 26$ .

Standard Form of a Line

The standard form of a line is given by $Ax + By = C$ , where $A$ , $B$ , and $C$ are constants.

Convert the Equation to Standard Form

We can convert the equation of the perpendicular line to standard form by multiplying both sides of the equation by $1$ .

from sympy import symbols, Eq, solve

# Define the variables
x, y = symbols('x y')

# Define the equation
eq = Eq(y, 5*x - 26)

# Multiply both sides by 1
eq = Eq(eq.lhs, 1*eq.rhs)

print(eq)

The output of the above code is $y = 5x - 26$ .

Final Answer

The final answer is $\boxed{y = 5x - 26}$ .

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Introduction

In our previous article, we discussed how to find the equation of a line that is perpendicular to a given line. In this article, we will answer some frequently asked questions related to this topic.

Q: What is the slope of the perpendicular line?

A: The slope of the perpendicular line is the negative reciprocal of the slope of the given line. If the slope of the given line is $m$ , then the slope of the perpendicular line is $-\frac{1}{m}$ .

Q: How do I find the equation of the perpendicular line?

A: To find the equation of the perpendicular line, you need to use the point-slope form of a line, which is given by $y - y_1 = m(x - x_1)$ , where $(x_1, y_1)$ is a point on the line and $m$ is the slope. You can then solve for $y$ to get the equation of the perpendicular line.

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

A: The point-slope form of a line is given by $y - y_1 = m(x - x_1)$ , where $(x_1, y_1)$ is a point on the line and $m$ is the slope.

Q: How do I convert the equation of the perpendicular line to standard form?

A: To convert the equation of the perpendicular line to standard form, you need to multiply both sides of the equation by $1$ . This will give you the equation in the form $Ax + By = C$ , where $A$ , $B$ , and $C$ are constants.

Q: What is the standard form of a line?

A: The standard form of a line is given by $Ax + By = C$ , where $A$ , $B$ , and $C$ are constants.

Q: Can I use the slope-intercept form to find the equation of the perpendicular line?

A: Yes, you can use the slope-intercept form to find the equation of the perpendicular line. However, you need to be careful when using this method, as it may lead to incorrect results if not done correctly.

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

A: To find the slope of the given line, you need to rewrite the equation of the line in slope-intercept form, which is given by $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.

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

A: The slope-intercept form of a line is given by $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.

Q: Can I use the point-slope form to find the equation of the given line?

A: No, you cannot use the point-slope form to find the equation of the given line. The point-slope form is used to find the equation of a line that passes through a given point and has a given slope.

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

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

Q: What is the equation of the given line?

A: The equation of the given line is $x + 5y = -2$ .

Q: Can I use the standard form to find the equation of the given line?

A: Yes, you can use the standard form to find the equation of the given line. However, you need to be careful when using this method, as it may lead to incorrect results if not done correctly.

Q: How do I find the equation of the perpendicular line in standard form?

A: To find the equation of the perpendicular line in standard form, you need to multiply both sides of the equation by $1$ . This will give you the equation in the form $Ax + By = C$ , where $A$ , $B$ , and $C$ are constants.

Q: What is the equation of the perpendicular line in standard form?

A: The equation of the perpendicular line in standard form is $5x - y = 26$ .

Conclusion

In this article, we answered some frequently asked questions related to finding the equation of a line that is perpendicular to a given line. We hope that this article has been helpful in clarifying any doubts you may have had on this topic.