The Equation For Line \[$ S \$\] Is Given As \[$ 2x + 3y = -18 \$\]. Line \[$ T \$\], Which Is Parallel To Line \[$ S \$\], Passes Through The Point \[$(-10, 5)\$\]. What Is The Equation Of Line \[$ T

Feb 28, 2025 by ADMIN 201 views

**The Equation of a Parallel Line: A Step-by-Step Guide**

Introduction

In mathematics, the concept of parallel lines is a fundamental idea that plays a crucial role in various branches of mathematics, including geometry and algebra. When two lines are parallel, they never intersect, and their slopes are equal. In this article, we will explore how to find the equation of a line that is parallel to a given line, passing through a specific point.

The Given Line

The equation of the given line, denoted as line ${$ s $}$, is ${$ 2x + 3y = -18 $}$. This is a linear equation in the form ${$ Ax + By = C $}$, where ${$ A = 2 $}$, ${$ B = 3 $}$, and ${$ C = -18 $}$. To find the slope of this line, we can rewrite the equation in the slope-intercept form, ${$ y = mx + b $}$, where ${$ m $}$ is the slope and ${$ b $}$ is the y-intercept.

Finding the Slope of the Given Line

To find the slope of the given line, we can rearrange the equation ${$ 2x + 3y = -18 $}$ to isolate ${$ y $}$. Subtracting ${$ 2x $}$ from both sides gives us ${$ 3y = -2x - 18 $}$. Dividing both sides by ${$ 3 $}$ yields ${$ y = -\frac{2}{3}x - 6 $}$. Therefore, the slope of the given line is ${$ -\frac{2}{3} $}$.

The Parallel Line

Since line ${$ t $}$ is parallel to line ${$ s $}$, it must have the same slope as line ${$ s $}$. Therefore, the slope of line ${$ t $}$ is also ${$ -\frac{2}{3} $}$. We are given that line ${$ t $}$ passes through the point ${$ (-10, 5) $}$. Using the point-slope form of a linear equation, ${$ y - y_1 = m(x - x_1) $}$, where ${$ (x_1, y_1) $}$ is the given point and ${$ m $}$ is the slope, we can find the equation of line ${$ t $}$.

Finding the Equation of the Parallel Line

Substituting the given point ${$ (-10, 5) $}$ and the slope ${$ -\frac2}{3} $}$ into the point-slope form, we get ${$ y - 5 = -\frac{2}{3}(x + 10) $}$. To simplify this equation, we can multiply both sides by ${$ 3 $}$ to eliminate the fraction, yielding ${$ 3(y - 5) = -2(x + 10) $}$. Expanding the left-hand side gives us ${$ 3y - 15 = -2x - 20 $}$. Adding ${$ 15 $}$ to both sides and adding ${$ 2x $}$ to both sides yields ${$ 3y = -2x + 5 $}$. Dividing both sides by ${$ 3 $}$ gives us the final equation of line ${$ t $}$ ${$ y = -\frac{2{3}x + \frac{5}{3} $}$.

Conclusion

In this article, we have demonstrated how to find the equation of a line that is parallel to a given line, passing through a specific point. We started with the equation of the given line, ${$ 2x + 3y = -18 $}$, and found its slope to be ${$ -\frac{2}{3} $}$. We then used the point-slope form of a linear equation to find the equation of the parallel line, ${$ y = -\frac{2}{3}x + \frac{5}{3} $}$. This process can be applied to any pair of parallel lines, and it is an essential tool in various branches of mathematics.

Applications of Parallel Lines

Parallel lines have numerous applications in mathematics, science, and engineering. In geometry, parallel lines are used to define congruent angles and similar triangles. In algebra, parallel lines are used to solve systems of linear equations. In physics, parallel lines are used to describe the motion of objects in a straight line. In engineering, parallel lines are used to design and build structures such as bridges and buildings.

Real-World Examples of Parallel Lines

Parallel lines can be found in various real-world situations. For example, railroad tracks are parallel lines that run alongside each other. The lines of a grid on a piece of graph paper are also parallel lines. In architecture, parallel lines are used to design and build buildings and bridges. In art, parallel lines are used to create perspective and depth in a painting or drawing.

Conclusion

In conclusion, the equation of a parallel line can be found using the point-slope form of a linear equation. By substituting the given point and the slope into the point-slope form, we can find the equation of the parallel line. This process is essential in various branches of mathematics, science, and engineering, and it has numerous real-world applications.

Q: What is the equation of a parallel line?

A: The equation of a parallel line is a linear equation that has the same slope as the given line, but a different y-intercept. The equation of a parallel line can be found using the point-slope form of a linear equation.

Q: How do I find the equation of a parallel line?

A: To find the equation of a parallel line, you need to know the slope of the given line and a point that the parallel line passes through. You can use the point-slope form of a linear equation to find the equation of the parallel line.

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

A: The point-slope form of a linear equation is ${$ y - y_1 = m(x - x_1) $}$, where ${$ (x_1, y_1) $}$ is the given point and ${$ m $}$ is the slope.

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

A: To use the point-slope form to find the equation of a parallel line, substitute the given point and the slope into the point-slope form. Then, simplify the equation to find the final equation of the parallel line.

Q: What is the slope of a parallel line?

A: The slope of a parallel line is the same as the slope of the given line. If the slope of the given line is ${$ m $}$, then the slope of the parallel line is also ${$ m $}$.

Q: How do I find the equation of a parallel line if I only know the slope and a point?

A: If you only know the slope and a point, you can use the point-slope form to find the equation of the parallel line. Substitute the slope and the point into the point-slope form, and simplify the equation to find the final equation of the parallel line.

Q: Can I find the equation of a parallel line if I only know the equation of the given line?

A: Yes, you can find the equation of a parallel line if you only know the equation of the given line. First, find the slope of the given line by rewriting the equation in the slope-intercept form. Then, use the point-slope form to find the equation of the parallel line.

Q: How do I know if two lines are parallel?

A: Two lines are parallel if they have the same slope and never intersect. You can check if two lines are parallel by finding their slopes and comparing them. If the slopes are the same, then the lines are parallel.

Q: Can I find the equation of a parallel line if I only know the y-intercept and a point?

A: Yes, you can find the equation of a parallel line if you only know the y-intercept and a point. First, find the slope of the given line by rewriting the equation in the slope-intercept form. Then, use the point-slope form to find the equation of the parallel line.

Q: How do I find the equation of a parallel line if I only know the equation of the given line in standard form?

A: If you only know the equation of the given line in standard form, you can rewrite the equation in the slope-intercept form to find the slope. Then, use the point-slope form to find the equation of the parallel line.

Q: Can I find the equation of a parallel line if I only know the equation of the given line in point-slope form?

A: Yes, you can find the equation of a parallel line if you only know the equation of the given line in point-slope form. First, rewrite the equation in the slope-intercept form to find the slope. Then, use the point-slope form to find the equation of the parallel line.

Q: How do I know if a line is parallel to a given line?

A: A line is parallel to a given line if it has the same slope and never intersects the given line. You can check if a line is parallel to a given line by finding its slope and comparing it to the slope of the given line. If the slopes are the same, then the lines are parallel.