Problem 19 - Calculus: One and Several Variables

Question

Find the slope and \(y\) -intercept. $$7 x-3 y+4=0$$

Step-by-Step Solution

Answer

The slope of the line is \(frac{7}{3}\) and the y-intercept is \(-frac{4}{3}\).

Step 1: Rearrange to obtain the y-intercept form

Starting with the given equation \(7x-3y+4=0\), first isolate \(y\) on one side of the equation. This involves adding \(3y\) to both sides of the equation and subtracting \(4\), resulting in \(3y=7x-4\). Then, divide every term by \(3\) to solve for \(y\), leading to the equation \(y= frac{7}{3}x - frac{4}{3}\). This is now in the slope-intercept form \(y=mx+c\).

Step 2: Identify the slope

In the equation obtained from step 1, \(y= frac{7}{3}x - frac{4}{3}\), the coefficient of \(x\) is the slope. Therefore in this equation, the slope \(m\) is \(frac{7}{3}\).

Step 3: Identify the y-intercept

In the equation \(y= frac{7}{3}x - frac{4}{3}\), the constant term is the y-intercept. Therefore in this equation, the y-intercept \(c\) is \(-frac{4}{3}\).

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form is crucial when studying linear equations in algebra. It provides a direct way to graph a line and understand its behavior. The slope-intercept form is represented by the equation \(y = mx + c\), where \(m\) represents the slope and \(c\) denotes the y-intercept of the line.

In simpler terms, the slope \(m\) tells us how steep a line is, while the y-intercept \(c\) indicates the point where the line crosses the y-axis. To convert a linear equation into this form, one must solve for \(y\) and rearrange terms so that \(y\) is by itself on one side of the equal sign, and everything else is on the other side.

For the given exercise, transforming the equation \(7x - 3y + 4 = 0\) to slope-intercept form showed that the line has a slope of \(\frac{7}{3}\) and crosses the y-axis at \(-\frac{4}{3}\). This format not only makes it simpler to graph the line but also to compare it with other lines by quickly assessing their slopes and y-intercepts.

Linear Equations

A linear equation represents a straight line on a two-dimensional plane and is a fundamental concept in algebra. It is typically written in the form \(Ax + By = C\), where \(A\), \(B\), and \(C\) are constants. These equations can model real-world phenomena such as motion, economics, and more.

Linear equations feature two variables, usually \(x\) and \(y\), and the graph of such an equation will always result in a line. One of the powers of algebra is to manipulate these equations to reveal useful attributes like the slope and y-intercept, which then can be used to graph the line or solve problems.

In our exercise example, converting the equation \(7x - 3y + 4 = 0\) into slope-intercept form is actually an application of skills in manipulating a linear equation. Recognizing and rearranging the elements to isolate \(y\) demonstrates a clear understanding of the relationship between the coefficients and the variables.

Algebra

Algebra is a branch of mathematics that deals with symbols and rules for manipulating these symbols; it is a unifying thread of almost all of mathematics. It allows for the formulation of relationships, structures, and equations, and provides a means to solve these mathematical problems.

In the realm of algebra, finding the slope and y-intercept from a linear equation like in our exercise is a practical example of applying algebraic principles. Solving for \(y\) and rearranging the equation into slope-intercept form involve skills such as combining like terms, using the distributive property, and understanding equality and inequalities.

The capacity to translate a standard linear equation into the slope-intercept form is not only essential in graphing lines but also serves as an excellent example of the utility of algebra in organizing and simplifying complex problems. It involves recognizing patterns and applying systematic techniques to unveil properties that are not readily apparent in the original equation.