Problem 24 - Algebra 2

Question

\(y\) varies directly with \(x\). If \(y=4\) when \(x=-2,\) find \(x\) when \(y=6\)

Step-by-Step Solution

Answer

When \(y = 6\), the value of \(x\) is \(-3\).

Step 1: Understanding direct variation

Direct variation between two variables means that as one variable changes, the other changes at a constant rate. This is expressed as the equation \(y = kx\), where \(k\) is the constant of variation.

Step 2: Calculating the constant of variation

We can find the constant of variation \(k\) by substituting the given values of \(y\) and \(x\) into the direct variation equation, \(y = kx\). With \(y=4\) and \(x=-2\), we get \(4 = k(-2)\). Solving for \(k\), we divide both sides by \(-2\) to obtain \(k = -2\).

Step 3: Finding the new value of \(x\)

Now that we have the constant \(k = -2\), we can find the new value of \(x\) when \(y = 6\) using the same variation equation. Substituting \(k\) and \(y\), we get \(6 = -2x\). Dividing both sides by \(-2\) gives us \(x = -3\).

Key Concepts

Constant of Variation

When learning about direct variation, it's essential to understand the constant of variation, symbolized as 'k' in algebra. This is the factor that relates both variables in a direct variation equation. In simple terms, it is the ratio of the dependent variable (usually y) to the independent variable (x).

Let's consider our exercise where we had the equation with two known values, which were that when y is 4, x is -2. To find 'k', you plug these numbers into the equation, which looks like this: \(y=kx\). That gives us \(4=k(-2)\). When solving for 'k', you should get \(k=-2\). This means for every unit x decreases, y decreases by 2 units. Understanding the value of 'k' is crucial because it remains constant and allows us to predict y for any given x in the relationship.

Proportional Relationships

Direct variation is a type of proportional relationship, one where two quantities increase or decrease at the same rate. If you were to graph a direct variation, you'd get a straight line that passes through the origin (0,0), signifying that if one variable is zero, the other must also be zero. This relationship is often represented as \(y = kx\) where 'k' is our constant of variation.

In the context of our problem, this proportional relationship means that if x were to be doubled, y would also double as long as 'k' remained the same. This is because the ratio between y and x is fixed. It's like a recipe; when you keep the proportion of ingredients the same, you reliably get the same taste regardless of the quantity you make.

Algebraic Equations

Understanding how to manipulate algebraic equations is key in solving problems involving direct variation. When we deal with direct variation, our main tool is the algebraic equation \(y = kx\). This equation allows us to plug in different values and solve for the unknown. It's like a formula where you need to find a missing piece, and knowing any two of the three quantities (y, k, x) will help you find the third.

For example, from our previous steps, knowing 'k' and 'y', we were tasked to find 'x'. The equation turned into \(6 = -2x\), and by manipulating this equation (dividing both sides by -2), we found that \(x = -3\). In algebra, such manipulations are standard and follow set rules to ensure that we get accurate results. Remember to perform the same operation on both sides of the equation to maintain equality.