Question

Write the equation of each line in general form. \(y\) intercept \(=2 ;\) perpendicular to \(4 x-3 y=7\)

Short answer

The equation of the line is \(3x + 4y = 8\).

Step-by-step solution

Write the equation of the given line

First, write down the equation of the line which we have the information about. The given line is in general form: \(4x - 3y = 7\).

Find the slope of the given line

Convert the equation of the given line to the slope-intercept form, which is \(y = mx + b\), to find the slope. For the given equation \(4x - 3y = 7\), add \(3y\) to both sides and then subtract \(7\) from both sides, followed by dividing by \(-3\) to isolate y. The slope of the line, \(m\), is the coefficient of \(x\) after converting to the y = mx + b form.

Determine the slope of the line perpendicular to the given line

If two lines are perpendicular, the slopes of those lines are negative reciprocals. If the slope of one line is \(m\), the slope of the line perpendicular to it will be \(-1/m\).

Write the equation of the line with the known slope and y-intercept

Using the slope from Step 3 and the given y-intercept, write the equation of the new line in point-slope form, \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \((x_1, y_1)\) is the point given by the y-intercept (0,2).

Convert to general form

Finally, convert the equation found in Step 4 to general form, \(Ax + By = C\). To convert from the point-slope form to the general form, distribute any multiplication, move all terms to one side of the equation and ensure all coefficients are integers.

Key concepts

General Form of a Line

Understanding the general form of a line is crucial when working with linear equations. The general form is typically expressed as \(Ax + By = C\), where \(A\), \(B\), and \(C\) are integers, and \(x\) and \(y\) are the variables representing the coordinates of a point on the line. This form is particularly useful for quickly identifying the relationship between two variables and is the standard format for many applications, including graphing and solving systems of equations.

To convert an equation to the general form, one would typically start from another form like the slope-intercept form and rearrange the terms. For instance, if you have an equation \(y = mx + b\), you would subtract \(mx\) from both sides to get \(y - mx = b\), and, if necessary, multiply through to clear any fractions, thus ensuring \(A\), \(B\), and \(C\) are integers.

For students, remembering to keep the \(A\) term positive is a helpful tip, and if the \(A\) term is negative, you can multiply the entire equation by -1 to maintain the general form convention.

Slope-Intercept Form

The slope-intercept form is another fundamental concept in understanding linear equations. It is depicted as \(y = mx + b\), where \(m\) is the slope of the line, and \(b\) is the y-intercept, the point where the line crosses the y-axis. This form makes it straightforward to graph a line quickly as the slope tells you the steepness and direction of the line, and the y-intercept gives a starting point on the graph.

When you're given an equation in general form, you can convert it to the slope-intercept form by solving for \(y\). Take, for example, the general form \(4x - 3y = 7\). By isolating \(y\), you get \(y = \frac{1}{3}x - \frac{7}{3}\), which reveals a slope of \(\frac{1}{3}\) and a y-intercept of \(\frac{7}{3}\).

This form is beneficial for understanding the behavior of a line. For instance, if \(m > 0\), the line ascends from left to right, and if \(m < 0\), it descends. If \(m = 0\), the line is horizontal, corresponding to a constant function.

Perpendicular Line Slope

The concept of perpendicular line slope is pivotal when dealing with geometry and coordinate graphing. Lines that are perpendicular to each other intersect at a right angle, and their slopes have a particular relationship: they are negative reciprocals of each other. If you have a line with slope \(m\), then the line perpendicular to it will have a slope of \(\frac{-1}{m}\).

For instance, if you’re dealing with a line with a slope of \(\frac{4}{3}\), a perpendicular line will have a slope of \(\frac{-3}{4}\). Remembering this relationship can help you not only to draw perpendicular lines but also to solve problems involving them. When writing the equation of a perpendicular line, you’ll use this new slope along with a given point to construct the equation in either point-slope form or your final desired form, such as general form or slope-intercept form.

This property is particularly useful in coordinate geometry, where understanding the relationship between perpendicular slopes is essential in calculating angles, constructing figures, and solving problems involving right angles.