Examples of Problem Solving Involving Writing Equations for Lines: A Step-by-Step Guide
When solving problems, especially those involving geometry and algebra, it is crucial to have a solid understanding of how to write and manipulate linear equations. This article will explore various problems that require the writing of equations for lines and provide step-by-step solutions. We will use examples to illustrate these concepts and ensure that you are well-equipped to tackle similar problems in the future.
Introduction to Writing Equations for Lines
Standard Form: ax by c
The standard form of a linear equation is ax by c, where a, b, and c are constants, and a and b are not both zero. This form is useful because it allows you to easily identify the slope and y-intercept of a line when rearranged.
Solving Problems Involving Parallel and Perpendicular Lines
One common type of problem is determining the equation of a line that is parallel or perpendicular to a given line. These problems often require the use of slope properties.
Example 1: Parallel Lines
Consider the line given by the equation 3x - 2y 6.
First, convert it into the standard form of a linear equation: ax by c. Here, we have 3x - 2y 6. Next, identify the slope of the line. You can do this by converting the equation into the slope-intercept form y mx b, where m is the slope. Starting with 3x - 2y 6, subtract 3x from both sides to get: (-2y -3x 6). Now, divide both sides by -2: (frac{-2y}{-2} frac{-3x 6}{-2}), which simplifies to (y frac{3}{2}x - 3). The slope (m) of the line is (frac{3}{2}). To find the equation of a line parallel to this one that passes through a specific point, you can use the point-slope form of a linear equation: (y - y_1 m(x - x_1)). Say we want a line that passes through the point (4, 5). Substitute (m frac{3}{2}), (x_1 4), and (y_1 5) into the point-slope form: (y - 5 frac{3}{2}(x - 4)). Simplify: (2y - 10 3x - 12), which becomes (3x - 2y 2).Example 2: Perpendicular Lines
Let's consider the line 3x - 3y 12.
First, convert it into the standard form: (3x - 3y 12). Divide both sides by 3: (x - y 4). The slope of this line is 1 because it can be written as (y x - 4). To find the equation of a line that is perpendicular to this one, remember that the slope of a perpendicular line is the negative reciprocal of the original slope. Therefore, the slope of the perpendicular line is -1. Using the point-slope form again, let’s find the equation of the line that passes through the point (-2, -6) with a slope of -1: (y - (-6) -1(x - (-2))). Simplify: (y 6 -x - 2), which becomes (x y -8).Finding Points of Intersection
Another important problem type involves finding the point of intersection of two lines. This is typically done by solving the system of linear equations formed by the two lines.
Example of Intersection:
Consider the two lines: 3x - 2y 6 and (3x - 3y 12).
First, solve the system of equations. From the first equation: (3x - 2y 6), isolate (y): (y frac{3x - 6}{2}). Substitute this expression for (y) into the second equation: (3x - 3(frac{3x - 6}{2}) 12). Multiply through by 2 to clear the fraction: (6x - 9x 18 24). Simplify: (-3x 18 24). Solve for (x): (-3x 6), so (x -2). Substitute (x -2) back into the first equation: (3(-2) - 2y 6). Calculate: (-6 - 2y 6), so (-2y 12), and (y -6). The point of intersection is ((-2, -6)).Conclusion
Writing and solving equations for lines is a fundamental skill in algebra and geometry. By understanding the standard form, parallel and perpendicular line properties, and point of intersection methods, you can effectively solve a variety of problems. Practice these techniques regularly to improve your problem-solving abilities and ensure that you are well-prepared for more complex equations and scenarios.
Key terms to remember are writing equations for lines, problem solving, and linear equations.