Question

Write the equation of each line in general form. \(y\) intercept \(=-5 ;\) perpendicular to \(y=3 x-4\)

Short answer

The equation of the line in general form is x + 3y + 15 = 0.

Step-by-step solution

Determine the slope of the given line

The slope-intercept form of a line is expressed as y = mx + b, where m is the slope and b is the y-intercept. For the given line y = 3x - 4, the slope is 3.

Find the slope of the line perpendicular to the given line

The slope of a line perpendicular to another is the negative reciprocal of the other line's slope. Therefore, the slope of the line perpendicular to y = 3x - 4 is -1/3.

Use the y-intercept to write the equation

The y-intercept is given as -5. So, the equation of the line with slope -1/3 and y-intercept -5 is y = (-1/3)x - 5.

Convert to the general form of the equation

The general form is Ax + By + C = 0. Multiply through by -3 to eliminate the fraction: -3y = x + 15. Re-arranging gives x + 3y + 15 = 0.

Key concepts

Slope-Intercept Form

Understanding the slope-intercept form is crucial for solving various problems involving straight lines. This form is represented as y = mx + b, where m indicates the slope of the line, which tells us how steep the line is, and b represents the y-intercept, the point where the line crosses the y-axis.

For example, if we have a line with the equation y = 3x - 4, we can see that the slope, m, is 3 and the y-intercept, b, is -4. This tells us that for every unit increase in x, y increases by 3 units, and the line will cross the y-axis at the point (0, -4).

Being able to interpret and manipulate this form is essential for graphing lines and understanding their properties. It also forms the basis for calculating and understanding more complex geometric topics, such as determining the equation of a line that is perpendicular to a given line, as seen in the referenced exercise.

Perpendicular Lines

When it comes to perpendicular lines, their slopes have a unique relationship: they are negative reciprocals of each other. In simpler terms, if one line's slope is a, the line perpendicular to it will have a slope of -1/a.

For instance, given a line with the slope of 3 (from the equation y = 3x - 4), a line perpendicular to this one would have a slope of -1/3. This is a key concept when working with coordinate geometry because it allows us to understand the orientation of lines relative to each other.

It's important to grasp this concept not only for academic purposes but also for real-world applications, such as in architecture and engineering, where the perpendicularity of structures can be essential.

Y-Intercept

The y-intercept of a line is another fundamental concept in algebra and geometry. It is the point where the line crosses the y-axis. This is represented by the value of b in the slope-intercept form y = mx + b. Simply put, it is the value of y when x = 0.

In our exercise, the given y-intercept is -5. Therefore, regardless of the line's slope or its direction, we know it will pass through point (0, -5) on the graph. The y-intercept is especially useful because it provides a starting point for graphing the line and also helps in understanding the line's equation.

When solving problems or graphing lines, it's beneficial to first plot the y-intercept and then use the slope to determine the direction and steepness of the line. This is also an essential step when converting an equation from slope-intercept form to general form, ensuring that you keep the coordinates of the y-intercept consistent throughout the process.