Problem 40 - Thinking Mathematically

Question

In Exercises 33-40, a. Put the equation in slope-intercept form by solving for \(y\). b. Identify the slope and the \(y\)-intercept. c. Use the slope and y-intercept to graph the line. \(5 x+3 y=15\)

Step-by-Step Solution

Answer

The equation in slope-intercept form is \(y = -\frac{5}{3}x + 5\), with a slope of \(-\frac{5}{3}\) and a y-intercept of \(5\).

Step 1: Solve for \(y\)

The given equation is \(5x + 3y = 15\). To put this into slope-intercept form, isolate \(y\) by subtracting \(5x\) from both sides and then dividing everything by \(3\). This yields the equation \(y = -\frac{5}{3}x + 5\).

Step 2: Identify the slope and the y-intercept

Looking at the obtained equation, the number accompanying \(-\frac{5}{3}x\) is the slope (that is, \(-\frac{5}{3}\)) and the number at the end (that is, \(5\)) is the y-intercept.

Step 3: Graph the line

Start off by plotting the y-intercept at \((0, 5)\) on the y-axis. Then, use the slope \(-\frac{5}{3}\) by moving 5 units down (because of the negative sign) and 3 units to the right on the graph paper to find the next point. Draw a line passing through these two points.

Key Concepts

Understanding Slope-Intercept Form

The slope-intercept form is a way of writing a linear equation so that you can easily identify the slope and the y-intercept of a line. It’s the classic 'y=mx+b' format, where 'm' represents the slope and 'b' represents the y-intercept.

For example, if we have the equation from our exercise, first we rearrange it to isolate y: we move the term involving x to the other side, and then we divide by the coefficient in front of y, which is 3.
This manipulation transforms our equation into the comfortable slope-intercept form, \[\begin{equation}y = -\frac{5}{3}x + 5\end{equation}\]
, which is much easier to work with for graphing and understanding the behaviour of the line.

Deciphering Slope

The slope of a line is a measure of its steepness and direction. In the slope-intercept form, the slope is denoted by the coefficient of x, which in the example is \[\begin{equation}-\frac{5}{3}\end{equation}\]
. This means that for every 3 units we move to the right on the graph, we'll need to move 5 units down since the slope is negative.

Why is it negative?

A negative slope indicates that the line is falling as it moves from left to right. The larger the fraction (or number), the steeper the line. In our specific case, the slope of \[\begin{equation}-\frac{5}{3}\end{equation}\]
indicates quite a steep fall as we move along the x-axis.

Locating the Y-Intercept

The y-intercept is where the line crosses the y-axis. It’s often referred to by its position \[\begin{equation}(0, b)\end{equation}\]
, where 'b' is a constant. In our text exercise, the y-intercept is 5, as seen from the equation in slope-intercept form, \[\begin{equation}y = -\frac{5}{3}x + 5\end{equation}\]
.

This gives us a starting point for the graph at \[\begin{equation}(0, 5)\end{equation}\]
. From there, you'd count along the y-axis to 5 and make your first plot point. Once you've plotted this intercept, the slope tells you how to find additional points to draw the entire line.