Problem 36 - Elementary Algebra

Question

For Problems 1-36, graph each linear equation. (Objective 2) $$ x=-3 y $$

Step-by-Step Solution

Answer

Graph the line using the slope \(-1/3\) and y-intercept \(0\).

Step 1: Rewrite the Equation

The given equation is already in terms of a linear relationship between the variables \( x \) and \( y \). It is given as \( x = -3y \). To plot this equation, it can be helpful to express it in terms of \( y \). Thus, we rewrite it as \( y = -\frac{1}{3}x \).

Step 2: Identify the Slope and Y-intercept

Now that the equation is in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept, we identify \( m = -\frac{1}{3} \) and \( b = 0 \). This means the line crosses the y-axis at the origin.

Step 3: Plot the Y-intercept

Start by plotting the y-intercept on the graph. Since the y-intercept \( b = 0 \), the line crosses the y-axis at the origin \((0, 0)\).

Step 4: Use the Slope to Find Another Point

The slope \( m = -\frac{1}{3} \) indicates that for each increase of 1 unit in \( x \), \( y \) decreases by \( \frac{1}{3} \) unit. From the y-intercept \((0, 0)\), move 3 units right along the x-axis to \((3, 0)\), and move 1 unit down along the y-axis to \((3, -1)\).

Step 5: Draw the Line

Plot the second point \((3, -1)\) on the graph. Draw a straight line through the points \((0, 0)\) and \((3, -1)\). This line represents the equation \( x = -3y \).

Step 6: Verify the Line

Ensure that other points such as \((-3, 1)\) fit the equation \( x = -3y \). Substituting \( y = 1 \) into \( x = -3y \), we get \( x = -3 \times 1 = -3 \), confirming the point lies on the line.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is a handy way to write the equation of a line. It is expressed as \( y = mx + b \), where:

• \( m \) stands for the slope.
• \( b \) is the y-intercept.

The slope \( m \) tells you how steep the line is. It describes how much \( y \) changes for a change in \( x \). If the slope is positive, the line will rise as you move right. If it's negative, the line will fall. The y-intercept \( b \) is where the line crosses the y-axis. This makes it easy to start graphing the line.
For the equation \( y = -\frac{1}{3}x \), the slope \( m = -\frac{1}{3} \) tells us the line decreases by \( \frac{1}{3} \) unit for every 1 unit increase in \( x \). The y-intercept \( b = 0 \) indicates that the line crosses the y-axis at the origin \((0, 0)\). This slope-intercept form is crucial for quickly understanding the direction and position of a line.

Plotting Points

Plotting points is an essential step in graphing a linear equation. To properly plot points, begin at the y-intercept and then use the slope to find additional points.
For our equation, the y-intercept is at \((0, 0)\). You start by marking this spot on the graph where the line will cross the y-axis. Next, use the slope \( m = -\frac{1}{3} \). Since the slope tells us for each increase of 1 in \( x \), \( y \) decreases by \( \frac{1}{3} \), we plot another point by moving:

• 3 units to the right, and
• 1 unit down.

This will take you to \((3, -1)\). Points like this help you construct the line. When you connect these points, a straight line gradually emerges. Plotting more points verifies if they satisfy the equation.

Linear Relationships

In mathematics, a linear relationship is a type of relationship showing a constant rate of change between two variables. Each change in \( x \) results in a proportional change in \( y \). This is why the graph of a linear equation is a straight line.
For a linear equation like \( x = -3y \), transforming it into \( y = -\frac{1}{3}x \) reveals its linear nature. The constant slope \( -\frac{1}{3} \) reflects the consistent decrease in \( y \) for each increase in \( x \). This consistent rate is what makes lines predictable and straightforward to graph.

• Linear relationships are easy to identify because of their straight line graph.
• They simplify modeling real-world situations like speed, cost, and other proportional changes.

Understanding linear relationships is fundamental for interpreting graphs, making predictions, and solving equations efficiently.