Problem 39 - A Graphical Approach to Precalculus with Limits

Question

Write each equation in the form \(y=m x+b .\) (A suggested window for a comprehensive graph of the equation is given. $$\begin{aligned} &-2 x+7 y=4\\\ &[-5,5] \text { by }[-5,5] \end{aligned}$$

Step-by-Step Solution

Answer

The slope-intercept form is \(y = \frac{2}{7}x + \frac{4}{7}\).

Step 1: Identify the Standard Form Equation

The provided equation is in the standard form: \[-2x + 7y = 4.\] Our task is to convert it to the slope-intercept form \(y = mx + b\) where \(m\) is the slope and \(b\) is the y-intercept.

Step 2: Isolate the y-term

Start by moving the \(x\) term to the other side of the equation to isolate the \(y\)-term. Add \(2x\) to both sides: \[7y = 2x + 4.\]

Step 3: Solve for y

Divide the entire equation by 7 to solve for \(y\): \[y = \frac{2}{7}x + \frac{4}{7}.\] This places the equation in the slope-intercept form \(y = mx + b\).

Step 4: Identify the slope and intercept

The equation \(y = \frac{2}{7}x + \frac{4}{7}\) is in the form \(y = mx + b\). Here, \(m = \frac{2}{7}\), which is the slope, and \(b = \frac{4}{7}\), which is the y-intercept.

Key Concepts

Standard Form Equation

The standard form of a linear equation is written as \(Ax + By = C\), where \(A\), \(B\), and \(C\) are real numbers, and \(A\) and \(B\) are not both zero. This form is quite useful in many mathematical contexts because it clearly displays the limitations and relationships between \(x\) and \(y\). However, it does not directly show the slope or y-intercept like the slope-intercept form.
To convert from standard form to the more intuitive slope-intercept form \(y = mx + b\), you'll want to isolate \(y\) on one side of the equation. This involves manipulating the equation by moving terms and possibly dividing by a coefficient, just as we did with \(-2x + 7y = 4\). The goal is to rearrange the equation until it fits the structure of \(y = mx + b\), where \(m\) will be the coefficient of \(x\), and \(b\) will be a constant representing the y-intercept.

Slope

The slope of a line, represented as \(m\) in the slope-intercept equation \(y = mx + b\), indicates the steepness and direction of the line on a graph. Mathematically, the slope is the ratio of the vertical change to the horizontal change between any two points on a line, often remembered as "rise over run."

• If \(m > 0\), the line ascends from left to right, indicating a positive slope.
• If \(m < 0\), the line descends from left to right, indicating a negative slope.
• If \(m = 0\), the line is horizontal, indicating no change in \(y\) as \(x\) changes.

In the equation \(y = \frac{2}{7}x + \frac{4}{7}\), the slope \(m\) is \(\frac{2}{7}\). This means that for every 7 units you move horizontally, the line will rise 2 units vertically. Knowing the slope helps in graphing the line and understanding the rate of change.

Y-Intercept

The y-intercept of a line is a key aspect of its graph. It represents the point where the line crosses the y-axis, and is denoted by \(b\) in the equation \(y = mx + b\). This occurs where \(x = 0\). The y-intercept is a crucial point because it gives us a starting point for drawing the line on a graph.
To find the y-intercept from the equation, set \(x = 0\) and solve for \(y\). For example, in the equation \(y = \frac{2}{7}x + \frac{4}{7}\), when \(x = 0\), we have \(y = \frac{4}{7}\). Hence, the y-intercept is \(\frac{4}{7}\), meaning the line intersects the y-axis at \(y = \frac{4}{7}\).
Understanding the y-intercept allows us to quickly visualize where our line will cut across the y-axis, making graph drawing more intuitive.