Problem 16 - Algebra 1

Question

Graph the equation. Find the constant of variation and the slope of the direct variation model. $$y=-5 x$$

Step-by-Step Solution

Answer

The constant of variation and the slope of the direct variation model is -5.

Step 1: Identify the constant of variation

In the given equation \(y = -5x\), the constant of variation is the coefficient before \(x\), which is -5.

Step 2: Identify the slope

The slope of a direct variation equation is the constant of variation. Hence, in this case, the slope is -5.

Step 3: Graph the equation

Plot the graph of the equation \(y = -5x\). Begin by graphing the y-intercept which is at the origin (0,0). Since the slope is -5, this means for every movement 1 unit to the right on the x-axis, we move 5 units downwards on the y-axis.

Key Concepts

Constant of Variation

In a direct variation equation, the constant of variation is a key number. It represents a fixed ratio between the variables involved. For the equation \(y = -5x\), the constant of variation is the number -5, which is also the coefficient of \(x\). This constant shows how much \(y\) changes in response to \(x\). If \(x\) increases by 1 unit, then \(y\) will decrease by 5 units because the relationship itself is negative.
This constant of variation determines the rate at which the variables interact. In direct variation, the relationship is such that an increase or decrease in one variable results in a proportional change in the other variable.

• It's crucial because it defines the specific relationship between \(x\) and \(y\).
• It is always a non-zero value, making it easy to spot in the equation.

Slope

The slope, a central concept in graphing linear equations, measures the steepness or tilt of a line. In the equation \(y = -5x\), the slope is -5. The slope determines how much \(y\) changes for a unit increase in \(x\).
Simply put, the slope is the constant of variation in direct variation equations. Here’s what the slope tells us:

• A slope of -5 means the line descends 5 units vertically for every unit it moves horizontally to the right.
• The negative sign indicates a downward tilt from left to right.

Understanding slope is vital to analyzing how quickly changes occur in linear relationships. It's what tells you how to move from point to point on the graph.

Graphing Linear Equations

Graphing linear equations like \(y = -5x\) allows us to visually interpret the relationship between variables. Start graphing by identifying key components: the y-intercept and the slope.
The y-intercept is where the line crosses the y-axis. For \(y = -5x\), it starts at the origin, or point (0,0). The slope gives directions on how to plot further points:

• From (0,0), move 1 unit to the right
• Then move 5 units downward, as per the slope of -5

Repeat these steps to draw the line. This process translates the algebraic expression into a visual form, making the data easier to understand and analyze. Ensure your graph extends across a range of \(x\) values to illustrate the complete relationship.

Algebra 1

Algebra 1 forms the foundation for understanding direct variation and graphs like \(y = -5x\). It introduces key concepts such as variables, equations, and graphing.
This level emphasizes the importance of expressing relationships clearly with equations and interpreting their meanings.

• Understanding direct variation helps make sense of real-world situations modeled by linear equations.
• It builds confidence in applying mathematical concepts to solve problems logically.

Mastering Algebra 1 skills aids in comprehending more complex math. These basics enable a smoother transition into Algebra 2 and beyond. Always focus on understanding how the parts of an equation work together to represent relationships visually and algebraically.