Problem 29 - Beginning and Intermediate Algebra

Question

Solve. If \(y\) varies jointly as \(x\) and \(z,\) and \(y=60\) when \(x=4\) and \(z=3,\) find \(y\) when \(x=7\) and \(z=2\)

Step-by-Step Solution

Answer

When \(x = 7\) and \(z = 2\), the value of \(y\) is 70.

Step 1: Understand Joint Variation Formula

Joint variation means that a variable varies with the product of two or more other variables. In this case, \(y\) varies jointly as \(x\) and \(z\). The formula for joint variation is \(y = kxz\), where \(k\) is the constant of variation.

Step 2: Substitute the Given Values and Solve for k

In the first scenario, the values are \(x = 4\), \(z = 3\), and \(y = 60\). We can substitute these values into the joint variation formula to solve for the constant of variation \(k\). \[60 = k(4)(3)\]

Step 3: Solve for k

The equation from step 2 simplifies to: \(60 = 12k\) Now, we divide both sides by 12 to solve for \(k\). \(k = \frac{60}{12}\) \(k = 5\)

Step 4: Substitute the k-value and the New x and z Values

Now that we have found the constant of variation \(k = 5\), we can find \(y\) in the second scenario, where \(x = 7\) and \(z = 2\). We'll use the joint variation formula with the given values and our constant of variation: \(y = (5)(7)(2)\)

Step 5: Calculate the New y-value

The equation from step 4 simplifies to: \(y = 5\times 7\times 2\) \(y = 70\) So, when \(x = 7\) and \(z = 2\), the value of \(y\) is 70.

Key Concepts

Constant of Variation

In joint variation problems, the relationship between variables involves a constant, known as the constant of variation, represented by the letter \(k\). This constant helps to express how one variable changes in response to changes in other variables. For joint variation, the formula we use is \(y = kxz\). Here, \(k\) shows how strongly \(y\) changes with respect to \(x\) and \(z\).
To determine \(k\), you need specific values of \(x\), \(y\), and \(z\). First, plug these values into the formula. For instance, if \(y = 60\), \(x = 4\), and \(z = 3\), you substitute these into the equation to find \(k\):

• \(60 = k \times 4 \times 3\)
• This simplifies to \(60 = 12k\)
• By dividing both sides by 12, you find \(k = 5\)

With \(k\) known, you can solve any related problem with new values of \(x\) and \(z\), using the same formula. This constant helps to consistently describe the joint variation under the same conditions.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They are used to represent mathematical relationships involving variables. In our joint variation problem, the key algebraic expression is \(y = kxz\). Here, the expression delineates a relationship between the variables \(y\), \(x\), and \(z\).
The expression is comprised of:

• Variables \(x\), \(y\), and \(z\) representing numbers that can change or take on different values.
• A constant \(k\) which helps define the specific relationship in this joint variation scenario.
• Operations include multiplication, highlighting how each part of the expression influences the overall relationship.

When algebraic expressions involve multiple variables, like in joint variation, they form equations that need solving. By understanding how to manipulate these expressions, you gain insights into predicting or explaining one variable through others.

Solving Equations

Solving equations in joint variation involves several straightforward steps. The goal is to find the unknown variable by expressing it in terms of known quantities. Following these steps helps you decipher problems more efficiently.
To solve the exercise problem about joint variation, start by identifying the given values. In the initial case:

• \(y = 60\)
• \(x = 4\)
• \(z = 3\)

Substitute these values into the joint variation equation \(y = kxz\) to find \(k\). Next, simplify and solve for \(k\), like so:

• \(60 = k \times 4 \times 3\)
• Simplifies to \(60 = 12k\)
• Now divide, \(k = \frac{60}{12}\)
• So, \(k = 5\).

Once \(k\) is known, use it to find \(y\) for new \(x\) and \(z\) values, such as \(x = 7\) and \(z = 2\):

• \(y = 5 \times 7 \times 2\)
• Simplified to \(y = 70\)

Following these steps ensures you accurately solve equations arising from joint variation scenarios, reinforcing your algebraic problem-solving skills.