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Mobile App Chapter 5: Problem 22
If \(x=y^2, y=4 z\) and \(x y z=16, x\) is: A. 16 B. 4 C. 8 D. 6 E. 2
Short Answer
Expert verified
B. 4
Step by step solution
01
Expressing variables in terms of others
Given: \(x = y^2\) and \(y = 4z\). Substitute \(y = 4z\) into the equation \(x = y^2\).
02
Substitute and simplify
Substitute \(y = 4z\) into \(x = y^2\): \(x = (4z)^2\). Therefore, \(x = 16z^2\).
03
Use the given product equation
Given: \(xyz = 16\). Substitute \(x = 16z^2\) and \(y = 4z\) from previous steps into this product equation.
04
Substitute and solve for \(z\)
Substitute into \(xyz = 16\): \(16z^2 \times 4z \times z = 16\). Simplify this to \(64z^4 = 16\).
05
Solve for \(z\)
Divide both sides by 64: \(z^4 = \frac{16}{64} = \frac{1}{4}\). Taking the fourth root of both sides, \(z = \frac{1}{\root 4 \relax 4} = \frac{1}{\root 4 \relax 2^2} = \frac{1}{\root 4 \relax 4} = \frac{1}{\root 4 \relax 4}= \frac{1}{\root 4 16}= \frac{1}{4}\root 4 (2^2)=\frac{1}{2}\).
06
Calculate \(x\)
Substitute \(z\) back into \(x = 16z^2\). With \(z = 0.5\), we have \(x = 16(0.5)^2 = 16 \times 0.25 = 4\).
07
Conclusion
Therefore, \(x\) is equal to 4.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
algebra
Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. In this problem, algebra is used to express relationships between different variables. By understanding the rules of algebra, we can translate word problems into mathematical equations and solve them step-by-step. Here, the given conditions are: \(x = y^2\), \(y = 4z\), and \(xyz = 16\). By expressing one variable in terms of others, we can simplify and solve the given relationships systematically.
variable substitution
Variable substitution is a key technique in algebra, where we replace one variable with another expression in terms of a different variable. This helps in simplifying complex equations and finding solutions. In this exercise, we had the following relationships: \(x = y^2\) and \(y = 4z\). By substituting \(y = 4z\) into \(x = y^2\), we transformed the equation into \(x = (4z)^2 = 16z^2\). This substitution made it easier to combine equations and eventually solve for the variables.
equation solving
Equation solving is the process of finding the values of variables that satisfy given mathematical equations. In our problem, after substitution, we ended up with \(xyz = 16\). Using the substitutions \(x = 16z^2\) and \(y = 4z\), we rewrote the equation as: \(16z^2 \times 4z \times z = 16\), which simplifies to \(64z^4 = 16\). Solving this equation required dividing both sides by 64 and taking the fourth root, giving \(z = 0.5\). Finally, we used this value to find \(x\).
mathematical simplification
Mathematical simplification involves reducing an equation or expression to its simplest form, making it easier to solve. In the solution provided, several steps of simplification were needed: starting from \(64z^4 = 16\), we divided both sides by 64 to get \(z^4 = \frac{1}{4}\). Taking the fourth root of both sides yielded \(z = 0.5\). Plugging this back into \(x = 16z^2\) simplified it to \(x = 16 \times (0.5)^2 = 4\). Thus, through systematic simplification, we deduced that \(x = 4\).
