Problem 27 - Algebra 2

Question

\(y\) varies directly with \(x\). If \(y=5\) when \(x=-3,\) find \(y\) when \(x=-1\)

Step-by-Step Solution

Answer

y=5/3 when x=-1.

Step 1: Identify the type of variation

The problem states that 'y varies directly with x', which indicates that the relationship between y and x can be described by the equation y = kx, where k is the constant of variation.

Step 2: Determine the constant of variation, k

Using the given values y=5 when x=-3, substitute them into the direct variation equation to find the constant k. The equation becomes 5 = k(-3). Solve for k by dividing both sides of the equation by -3, resulting in k = -5/3.

Step 3: Write the direct variation equation with the found constant

Substitute the value of k into the direct variation equation, y = kx, giving us y = (-5/3)x.

Step 4: Find y for the given value of x

Substitute x = -1 into the equation y = (-5/3)x to find the corresponding value of y. This gives us y = (-5/3)(-1), which simplifies to y = 5/3.

Key Concepts

Understanding Direct Variation

When talking about direct variation, we mean that two variables increase or decrease together at a consistent rate. In algebra, this relationship is expressed as y = kx, where y and x are the variables and k is a non-zero constant known as the constant of variation. The equation represents a straight line passing through the origin (0,0) when graphed.

For a student grappling with this concept, it's essential to understand that as one variable changes, the other changes in a proportional manner. If x doubles, y also doubles if they are in direct variation. An easy way to remember direct variation is the phrase 'y varies directly with x', which implies that any change in x will cause a predictable change in y based on the constant k.

The Role of Constant of Variation

The constant of variation, often denoted as k, plays a pivotal role in the relationship described by a direct variation. It is the consistent rate at which the variables change together. Determining the k is often the first step when solving direct variation problems because it sets up the equation needed to find other variable values.

To find k, you can use the given values of the variables x and y. In our example, we used the given information that y = 5 when x = -3 and discovered that k = -5/3. With this constant, we can predict y for any value of x that varies directly with it. This constant is what keeps the relationship between x and y consistent and predictable.

Algebraic Equations in Direct Variation

In the context of algebraic equations and direct variation, the equation y = kx provides a foundation for manipulating and solving problems. Algebraic equations are statements of equality that contain variables and constants and can be simplified or rearranged to find unknown values.

When working with direct variation, if you're already given k, you can substitute it along with any value for x to find the corresponding y. This substitution method transforms the general equation into a specific solution for the variables. In our textbook problem, we used the algebraic equation with the found constant of variation to determine y for x = -1, resulting in y = 5/3. This step illustrates how algebraic manipulation can provide clear solutions to direct variation problems, allowing for a deeper understanding and application in various mathematical scenarios.