Problem 25 - Algebra 2

Question

If \(y\) varies directly as \(x\) and \(y=9\) when \(x\) is \(-15,\) find \(y\) when \(x=21\)

Step-by-Step Solution

Answer

When x = 21, y is -12.6.

Step 1: Understand the Relationship

Since y varies directly as x, it means that y = kx for some constant k. This formula describes a direct variation, meaning y and x increase or decrease together.

Step 2: Find the Constant of Variation

Substitute the given values y = 9 and x = -15 into the direct variation equation y = kx to find k.\[ 9 = k(-15) \] Solve for k to get: \[ k = \frac{9}{-15} = -\frac{3}{5} \] This is the constant of proportionality or variation.

Step 3: Use the Constant to Find New y Value

Now that you have k = -\frac{3}{5}, use it to find y when x = 21. Substitute x = 21 into the equation y = kx.\[ y = -\frac{3}{5} \times 21 \] Calculate to find y: \[ y = -\frac{63}{5} = -12.6 \] This is the value of y when x = 21.

Key Concepts

Constant of Proportionality

The Constant of Proportionality, often represented by the symbol \( k \), is a crucial concept in direct variation. Whenever two quantities vary directly, they are connected by this constant. This means that if one quantity doubles, the other will double if \( k \) remains unchanged. It's like a fixed multiplier that connects the two variables. In the context of the problem, \( y \) and \( x \) are directly proportional. We found the constant of proportionality by substituting the given values into the equation \( y = kx \). Inserting \( y = 9 \) and \( x = -15 \), the calculation \( 9 = k(-15) \) is made. Solving this gives \( k = -\frac{3}{5} \). Once you find \( k \), you can use it to explore how \( y \) changes as \( x \) changes.

Direct Variation Equation

A Direct Variation Equation is a simple mathematical representation that expresses how two quantities change in relation to each other. In this relation, one variable is always equal to some constant multiplied by the other variable. This is expressed as: \[ y = kx \] Key points about this equation:

• The constant \( k \) determines how steep or flat the relationship is between \( y \) and \( x \).
• Both variables will increase or decrease together in a linear fashion if \( k \) is positive and inversely if \( k \) is negative.
• The graph of a direct variation is always a straight line passing through the origin if plotted in a coordinate plane.

In the problem, by identifying \( k = -\frac{3}{5} \), the equation becomes \( y = -\frac{3}{5}x \). This allows us to determine any \( y \) for a given \( x \).

Solving for Variables in Direct Proportion

When you know the Direct Variation Equation and the constant of proportionality, you can solve for missing values effectively. Here's how to handle such problems with ease:If you want to find \( y \) for a new \( x \), just replace \( x \) in the equation with its new value. From our specific case, now that we have found \( k = -\frac{3}{5} \), all we need is the equation \( y = kx \). To find \( y \) when \( x = 21 \), the substitution becomes: \[ y = -\frac{3}{5} \times 21 \] Calculate it step by step:

• Multiply \( -\frac{3}{5} \) by \( 21 \), which results in \( y = -\frac{63}{5} \).
• Simplify to get \( y = -12.6 \).

Thus, for \( x = 21 \), \( y \) becomes \( -12.6 \). This straightforward process emphasizes how the value of one variable affects the other in direct proportion scenarios.