Problem 33 - 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. \(3 x+y=0\)

Step-by-Step Solution

Answer

The slope-intercept form is \(y=-3x\). The slope is -3 and the y-intercept is 0. To graph the line, start at the y-intercept and, apply slope (down 3 units, right 1 unit) to draw the line.

Step 1: Convert to slope-intercept

To convert \(3x+y=0\) into slope-intercept form, isolate \(y\). This is done by subtracting \(3x\) from both sides, which gives \(y=-3x\). So, the equation in slope-intercept form is \(y = -3x\).

Step 2: Identify slope and y-intercept

In the equation \(y=-3x\), the number multiplying \(x\) is -3, this is our slope \(m\), so \(m=-3\). There is no constant added to \(-3x\), which implies that our y-intercept \(b=0\). Hence, the slope is -3 and the y-intercept is 0.

Step 3: Graph the line

For the graph, start by plotting the y-intercept (0,0). The slope is -3, which can be considered as -3/1 and this implies to move 3 units down (since it's negative) and 1 unit to the right (since it's positive), starting from y-intercept. Do these steps couple of times and draw a line through the points.

Key Concepts

Identifying slope

The slope of a line is a measure of its steepness and direction. It is usually represented by the letter \(m\) in the equation. The slope is calculated as the "rise over run," or the change in \(y\) divided by the change in \(x\) between two points on the line. In simpler terms, the slope tells us how much \(y\) changes for a unit change in \(x\).
For the equation \(y = -3x\), the slope is \(-3\). Here, \(-3\) means that for every 1 unit increase in \(x\), \(y\) decreases by 3 units. This indicates a downward slanting line. The slope can be visualized or calculated by identifying two points on a graph, calculating the difference in the \(y\)-values (rise), and dividing by the difference in \(x\)-values (run).

• Positive slope: Line rises as it moves from left to right.
• Negative slope: Line falls as it moves from left to right.
• Zero slope: Horizontal line, no rise or fall.
• Undefined slope: Vertical line, no run.

y-intercept

The \(y\)-intercept of a line is the point where it crosses the \(y\)-axis. This point is crucial as it gives one point of the line without needing to perform calculations. In the slope-intercept form \(y = mx + b\), the \(b\) represents the \(y\)-intercept.
For our equation \(y = -3x\), we notice there's no number added to \(-3x\), thus \(b = 0\). This means the line crosses the \(y\)-axis at the origin, which is the point (0,0).
Understanding the \(y\)-intercept helps when graphing because it gives us a starting point for the line on the graph. Regardless of the slope, each line will cross the \(y\)-axis at the intercept specified in the equation.

Graphing linear equations

Graphing linear equations involves plotting the \(y\)-intercept and using the slope to find other points on the line. Let's break this down:
1. **Start with the \(y\)-intercept**: Begin by plotting the \(y\)-intercept on the graph. For the equation \(y = -3x\), we start at the origin (0,0). This is our initial point.
2. **Use the slope to find more points**: The slope \(-3\) can be expressed as \(-3/1\). This fraction means that for each 1 unit you move to the right (positive \(x\)-direction), you move 3 units down (negative \(y\)-direction). Apply this repeatedly to identify multiple points. For example, from (0,0), move to (1,-3), and then to (2,-6).
3. **Draw the line**: Once you have plotted enough points using the slope, draw a line through the points. This line continues infinitely in both directions and represents all solutions to the equation.Graphing effectively requires understanding both the slope and the \(y\)-intercept, allowing you to plot accurate and meaningful graphs.