Find An Equation For The Line With The Given Property. Write The Equations In Slope-intercept Form.(a) Perpendicular To The Line X − 5 Y = 3 X - 5y = 3 X − 5 Y = 3 And Containing The Point ( 5 , 3 (5, 3 ( 5 , 3 ].(b) Parallel To The Line 2 X − 5 Y = 3 2x - 5y = 3 2 X − 5 Y = 3 And

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In this article, we will explore how to find the equation of a line with specific properties. We will focus on two main cases: finding the equation of a line that is perpendicular to a given line and contains a specific point, and finding the equation of a line that is parallel to a given line.

Case (a): Perpendicular to the Line $x - 5y = 3$ and Containing the Point $(5, 3)$

To find the equation of a line that is perpendicular to the given line and contains the point $(5, 3)$, we need to follow these steps:

Step 1: Convert the Given Line to Slope-Intercept Form

The given line is in the form $x - 5y = 3$. To convert it to slope-intercept form, we need to isolate $y$.

x - 5y = 3
-5y = -x + 3
y = \frac{x}{5} - \frac{3}{5}

Step 2: Find the Slope of the Given Line

The slope of the given line is the coefficient of $x$ in the slope-intercept form.

m = \frac{1}{5}

Step 3: Find the Slope of the Perpendicular Line

The slope of the perpendicular line is the negative reciprocal of the slope of the given line.

m' = -\frac{1}{\frac{1}{5}}
m' = -5

Step 4: Use the Point-Slope Form to Find the Equation of the Perpendicular 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.

y - 3 = -5(x - 5)
y - 3 = -5x + 25
y = -5x + 28

Case (b): Parallel to the Line $2x - 5y = 3$ and Containing the Point $(7, 2)$

To find the equation of a line that is parallel to the given line and contains the point $(7, 2)$, we need to follow these steps:

Step 1: Convert the Given Line to Slope-Intercept Form

The given line is in the form $2x - 5y = 3$. To convert it to slope-intercept form, we need to isolate $y$.

2x - 5y = 3
-5y = -2x + 3
y = \frac{2x}{5} - \frac{3}{5}

Step 2: Find the Slope of the Given Line

The slope of the given line is the coefficient of $x$ in the slope-intercept form.

m = \frac{2}{5}

Step 3: Find the Slope of the Parallel Line

The slope of the parallel line is the same as the slope of the given line.

m' = \frac{2}{5}

Step 4: Use the Point-Slope Form to Find the Equation of the Parallel 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.

y - 2 = \frac{2}{5}(x - 7)
y - 2 = \frac{2}{5}x - \frac{14}{5}
y = \frac{2}{5}x - \frac{14}{5} + 2
y = \frac{2}{5}x - \frac{14}{5} + \frac{10}{5}
y = \frac{2}{5}x - \frac{4}{5}

Conclusion

In this article, we have explored how to find the equation of a line with specific properties. We have found the equation of a line that is perpendicular to a given line and contains a specific point, and the equation of a line that is parallel to a given line and contains a specific point. We have used the point-slope form and the slope-intercept form to find the equations of the lines.

References

• [1] "Algebra" by Michael Artin
• [2] "Linear Algebra and Its Applications" by Gilbert Strang
• [3] "Calculus" by Michael Spivak

Further Reading

• [1] "Introduction to Linear Algebra" by Gilbert Strang
• [2] "Linear Algebra and Its Applications" by Gilbert Strang
• [3] "Calculus" by Michael Spivak

Tags

• mathematics
• algebra
• linear algebra
• calculus
• point-slope form
• slope-intercept form
• perpendicular lines
• parallel lines
• equations of lines
  

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In this article, we will answer some frequently asked questions about finding equations of lines. We will cover topics such as perpendicular lines, parallel lines, and the point-slope form.

Q: What is the difference between a perpendicular line and a parallel line?

A: A perpendicular line is a line that intersects another line at a 90-degree angle, while a parallel line is a line that never intersects another line and is always the same distance apart.

Q: How do I find the equation of a line that is perpendicular to a given line and contains a specific point?

A: To find the equation of a line that is perpendicular to a given line and contains a specific point, you need to follow these steps:

1. Convert the given line to slope-intercept form.
2. Find the slope of the given line.
3. Find the slope of the perpendicular line (which is the negative reciprocal of the slope of the given line).
4. Use the point-slope form to find the equation of the perpendicular line.

Q: How do I find the equation of a line that is parallel to a given line and contains a specific point?

A: To find the equation of a line that is parallel to a given line and contains a specific point, you need to follow these steps:

1. Convert the given line to slope-intercept form.
2. Find the slope of the given line.
3. Find the slope of the parallel line (which is the same as the slope of the given line).
4. Use the point-slope form to find the equation of the parallel 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 a line from slope-intercept form to point-slope form?

A: To convert a line from slope-intercept form to point-slope form, you need to isolate $y$ and then rewrite the equation in the form $y - y_1 = m(x - x_1)$.

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: How do I find the slope of a line?

A: To find the slope of a line, you need to look at the coefficient of $x$ in the slope-intercept form of the line.

Q: What is the y-intercept of a line?

A: The y-intercept of a line is the point where the line intersects the y-axis.

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

A: To find the equation of a line that passes through two points, you need to follow these steps:

1. Find the slope of the line using the two points.
2. Use the point-slope form to find the equation of the line.

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 follow these steps:

1. Find the slope of the line using the two points.
2. Use the point-slope form to find the equation of the line.

m = \frac{5 - 3}{4 - 2}
m = \frac{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 $(1, 2)$ and $(3, 4)$?

A: To find the equation of a line that passes through the points $(1, 2)$ and $(3, 4)$, you need to follow these steps:

1. Find the slope of the line using the two points.
2. Use the point-slope form to find the equation of the line.

m = \frac{4 - 2}{3 - 1}
m = \frac{2}{2}
m = 1
y - 2 = 1(x - 1)
y - 2 = x - 1
y = x - 1 + 2
y = x + 1

Conclusion

In this article, we have answered some frequently asked questions about finding equations of lines. We have covered topics such as perpendicular lines, parallel lines, and the point-slope form. We have also provided examples of how to find the equation of a line that passes through two points.

References

• [1] "Algebra" by Michael Artin
• [2] "Linear Algebra and Its Applications" by Gilbert Strang
• [3] "Calculus" by Michael Spivak

Further Reading

• [1] "Introduction to Linear Algebra" by Gilbert Strang
• [2] "Linear Algebra and Its Applications" by Gilbert Strang
• [3] "Calculus" by Michael Spivak

Tags

• mathematics
• algebra
• linear algebra
• calculus
• point-slope form
• slope-intercept form
• perpendicular lines
• parallel lines
• equations of lines