Problem 18 - Algebra 1

Question

Graph the equation. Find the constant of variation and the slope of the direct variation model. $$y=-x$$

Step-by-Step Solution

Answer

The constant of variation (k) is -1, and the slope of the direct variation model is -1.

Step 1: Identify the Constant of Variation

The constant of variation (k) in a direct variation model is the coefficient associated with the variable. In this case, the given equation is \(y=-x\). So, the constant of variation, \(k\), is -1.

Step 2: Identify the Slope

In the equation of a line, \(y=mx+ b\), where m is the slope and b is the y-intercept, the slope is the coefficient associated with the variable x. Comparing this with our given equation, we can see the coefficient of x is -1. Hence the slope is -1.

Step 3: Graph the Equation

The given equation can be graphed on a graph paper by choosing any x values and getting the corresponding y values using the equation \(y=-x\) (remember that y always equals the opposite of x for this equation). Plot these points on the graph paper and draw a line connecting them. The line will have a slope of -1 and y-intercept of 0.

Key Concepts

Graphing Equations

Graphing an equation involves plotting points that satisfy the equation and then connecting them to form a line. The process starts by selecting values for one of the variables, commonly starting with simple numbers like -2, -1, 0, 1, and 2 for clarity.
To graph the equation \( y = -x \), you choose values for \( x \) and compute the corresponding \( y \). For instance:

• If \( x = -2 \), then \( y = -(-2) = 2 \)
• If \( x = -1 \), then \( y = -(-1) = 1 \)
• If \( x = 0 \), then \( y = -(0) = 0 \)
• If \( x = 1 \), then \( y = -(1) = -1 \)
• If \( x = 2 \), then \( y = -(2) = -2 \)

You then plot these points on a coordinate graph and draw a straight line through them. This line illustrates the equation, showing how \( y \) changes inversely with \( x \). The graph of \( y = -x \) results in a line that slopes downward from top left to bottom right.

Slope

The slope of an equation represents how steep the line is, indicating how much \( y \) changes when \( x \) changes by one unit. In the linear equation format \( y = mx + b \), the slope is the coefficient \( m \).
For the equation \( y = -x \), the coefficient of \( x \) is -1. This implies that the slope is -1, meaning for every unit increase in \( x \), \( y \) decreases by one unit. Hence, the line goes downward. A negative slope moves left as it descends on the graph, while a positive slope would rise.
Slopes are a crucial aspect because they determine the direction and angle of the line. Observing the equation's slope, we can predict and understand the graphic presentation of a line without plotting additional points.

Constant of Variation

In direct variation, the constant of variation, denoted as \( k \), is a pivotal number that connects the two variables directly. In an equation \( y = kx \), the constant \( k \) is equivalent to the slope of the line.
With \( y = -x \), the constant of variation \( k \) is -1. This tells us that \( y \) varies directly with \( x \) but inversely due to the negative sign. As \( x \) increases, \( y \) decreases by an equivalent rate and vice versa.
The constant of variation plays a key role as it dictates the rate and direction at which one variable changes in relation to another. It facilitates the straightforward relationship between variables where the ratio of \( y \) to \( x \) remains constant.