Question

The equations of two lines are given. Determine if lines \(L_{1}\) and \(L_{2}\) are parallel, perpendicular, or neither. \(L_{1}: y=3 x+4 ; L_{2}: y=x-\frac{1}{4}\)

Short answer

Lines \(L_1\) and \(L_2\) are neither parallel nor perpendicular.

Step-by-step solution

Identify the Slopes

From the given equations, line \(L_{1}\) has a slope of 3 and line \(L_{2}\) has a slope of 1.

Compare the Slopes

Since the slopes of \(L_{1}\) and \(L_{2}\) are not equal, they are not parallel. If the product of the slopes was -1 they would be perpendicular. However, in our case, the product is \(3*1 = 3\), not -1, meaning the lines are also not perpendicular.

Final Conclusion

Since our lines are neither parallel nor perpendicular, the final conclusion is that the lines \(L_{1}\) and \(L_{2}\) are neither parallel nor perpendicular.

Key concepts

Slopes of Lines

Understanding the slope of a line in algebra is fundamental to graphing and solving linear equations. The slope is a measure of how steep a line is and is represented by the letter 'm'. When looking at the equation of a line in the slope-intercept form, which is written as \( y = mx + b \), the slope is the coefficient of 'x'.

For example, in the line equation \( y = 3x + 4 \), the slope is 3. This means for every one unit you move horizontally to the right, the line rises 3 units vertically. A positive slope means the line ascends from left to right, while a negative slope indicates it descends. If the slope is zero, the line is horizontal, and if the slope is undefined (which happens when the line is vertical), we cannot represent it as a number.

Parallel and Perpendicular Lines

Two lines are considered parallel when they have the same slope and will never intersect. On a graph, they look like train tracks, running alongside each other forever.

For example, the lines \( y = 2x + 1 \) and \( y = 2x - 5 \) are parallel because their slopes are both 2. On the flip side, two lines are perpendicular when their slopes are negative reciprocals of each other, which mathematically means their slopes multiply to -1. For instance, if one line has a slope of 4, a perpendicular line would have a slope of \( -\frac{1}{4} \). This relationship ensures that perpendicular lines intersect to form right angles.

Equations of Lines

The equation of a line is a powerful tool that describes its precise trajectory on a coordinate plane. This equation can take several forms, but the most common is the slope-intercept form: \( y = mx + b \), where 'm' represents the slope and 'b' signifies the y-intercept, the point where the line crosses the y-axis.

Other forms include the point-slope form, \( y - y_1 = m(x - x_1) \), useful when you know a point \( (x_1, y_1) \) on the line and its slope, and the standard form, \( Ax + By = C \), which is favored for theoretical analysis and can easily represent vertical lines, which the slope-intercept form cannot.

Algebraic Problem Solving

When approaching algebraic problem solving, especially with line equations, there are several key steps to follow. Start by clearly identifying what is known and what needs to be found—like in our example problem, we wanted to determine the relationship between two given lines.

Subsequently, extract key information from the equations provided, such as slopes and intercepts. Comparing these characteristics can reveal whether the lines are parallel, perpendicular, or neither. These problem-solving skills are not just limited to algebra; they can be applied to a wide array of mathematical problems, requiring logical thinking and step-by-step analysis to derive a solution.