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Chapter 7: Problem 14

Determine whether each equation represents direct, inverse, joint, or combined variation. $$ y=6 x^{3} z^{2} $$

Short Answer

Expert verified
Joint variation

Step by step solution

01

Identify the Variables

Observe and list the variables in the given equation. Here, the variables are y, x, and z.
02

Analyze the Equation Structure

Notice that y is expressed as a product involving both x and z. The equation is in the form of a multiplication involving variables.
03

Determine the Type of Variation

Since the equation y = 6x鲁z虏 involves multiple variables and each variable is raised to a power, it suggests a joint variation. Joint variation occurs when a variable is directly related to the product of two or more other variables, each raised to a power.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Variable Identification
The first step in understanding an algebraic equation is identifying its variables. Variables are symbols that represent quantities in mathematical expressions or equations. In the given equation, y = 6x鲁z虏 y, x, and z are the variables. Here:
  • y represents the dependent variable, which depends on the values of x and z.
  • x and z are independent variables, meaning they can change freely.
Understanding which symbols are variables helps in understanding their relationships. If you misidentify the variables, it might lead to incorrect interpretations and solutions.
Analyzing Equation Structure
After identifying the variables, the next step is to analyze the structure of the equation. In y = 6x鲁z虏y is expressed as a product of x鲁 and z虏, multiplied by the constant 6. Breaking it down, we see:
  • y is on one side of the equation, indicating it depends on the other side.
  • The other side has a constant (6) and two variables each raised to a power (x鲁 and z虏).
Observing this structure tells us a lot about how y changes with respect to x and z. Any time you see an equation like this, it鈥檚 key to recognize the general form and how the variables interact with each other.
Types of Variation in Algebra
Algebra often involves understanding how variables are related through different types of variation. Here are the most common types of variation:
  • Direct Variation: When one variable increases, the other variable also increases proportionally. Represented as y = kx.
  • Inverse Variation: When one variable increases, the other variable decreases proportionally. Represented as y = k / x.
  • Joint Variation: When a variable is directly proportional to the product of two or more other variables. Represented as y = kx鈦縵岬�.
  • Combined Variation: A combination of direct, inverse, and joint variations in one equation.
In our equation y = 6x鲁z虏, y varies jointly with x and z since y is directly related to the product of x and z, each raised to a power. This understanding is crucial for solving and interpreting problems involving such equations.

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Most popular questions from this chapter

Forensic scientists use the lengths of certain bones to calculate the height of a person. Two bones often used are the tibia \((t),\) the bone from the ankle to the knee, and the femur \((r),\) the bone from the knee to the hip socket. A person's height ( \(h\) ) in centimeters is determined from the lengths of these bones by using functions defined by the following formulas. For men: \(h(r)=69.09+2.24 r\) or \(h(t)=81.69+2.39 t\) For women: \(h(r)=61.41+2.32 r\) or \(h(t)=72.57+2.53 t\) (a) Find the height of a man with a femur measuring \(56 \mathrm{cm}\). (b) Find the height of a man with a tibia measuring \(40 \mathrm{cm} .\) (c) Find the height of a woman with a femur measuring \(50 \mathrm{cm} .\) (d) Find the height of a woman with a tibia measuring \(36 \mathrm{cm}\).

For Exercises \(47-58,(a)\) find the slope of the line through each pair of points, if possible, and (b) based on the slope, indicate whether the line through the points rises from left to right, falls from left to right, is horizontal, or is vertical. See Example 6 and FlGURE \(19 .\) $$(1.5,2.6) \text { and }(0.5,3.6)$$

Let \(f(x)=-3 x+4\) and \(g(x)=-x^{2}+4 x+1 .\) Find the following. $$ f(x+h)-f(x) $$

Let \(f(x)=-3 x+4\) and \(g(x)=-x^{2}+4 x+1 .\) Find the following. $$ f(x-2) $$

Solve each problem. f varies jointly as \(g^{2}\) and \(h,\) and \(f=50\) when \(g=4\) and \(h=2 .\) Find \(f\) when \(g=3\) and \(h=6\)
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