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Magazine Chapter 9: Problem 34
Find all solutions of the system of equations. $$\left\\{\begin{aligned}x^{4}-y^{3} &=17 \\\3 x^{4}+5 y^{3} &=53\end{aligned}\right.$$
Short Answer
Expert verified
\((x, y) = \left( \pm \sqrt[4]{\frac{69}{4}}, \sqrt[3]{\frac{1}{4}} \right) \)."
Step by step solution
01
Understand the System of Equations
We have two equations: (1) \( x^4 - y^3 = 17 \) and (2) \( 3x^4 + 5y^3 = 53 \). We are looking to find values of \( x \) and \( y \) that satisfy both equations simultaneously.
02
Solve Equation (1) for y
From Equation (1) \( x^4 - y^3 = 17 \), rewrite to express \( y^3 \): \( y^3 = x^4 - 17 \).
03
Substitute in Equation (2)
Substitute \( y^3 = x^4 - 17 \) into Equation (2). The equation becomes \( 3x^4 + 5(x^4 - 17) = 53 \).
04
Simplify the Equation
Distribute the 5 and simplify: \[ 3x^4 + 5x^4 - 85 = 53 \].
05
Combine Like Terms
Combine like terms in the equation: \( 8x^4 - 85 = 53 \).
06
Solve for x
Add 85 to both sides to solve for \( x^4 \): \( 8x^4 = 138 \). Now divide by 8: \( x^4 = \frac{138}{8} \). Simplify this to get \( x^4 = \frac{69}{4} \).
07
Solve for x Explicitly
Find \( x \) by taking the fourth root of \( \frac{69}{4} \): \( x = \pm \sqrt[4]{\frac{69}{4}} \).
08
Solve for y Using x
Substitute \( x = \pm \sqrt[4]{\frac{69}{4}} \) back into \( y^3 = x^4 - 17 \) to find values for \( y \): since \( x^4 = \frac{69}{4} \), \( y^3 = \frac{69}{4} - 17 \). Thus, \( y^3 = \frac{69}{4} - \frac{68}{4} = \frac{1}{4} \). So \( y = \sqrt[3]{\frac{1}{4}} \).
09
Combine Solutions
The solutions for the system of equations are \( (x, y) = \left( \pm \sqrt[4]{\frac{69}{4}}, \sqrt[3]{\frac{1}{4}} \right) \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Equation Solving
Solving a system of equations means finding the values of the variables that satisfy all equations involved. In this case, we have two equations that relate the variables \( x \) and \( y \). The goal is to find particular values of \( x \) and \( y \) that make both equations true simultaneously.
To solve a system, it often helps to express one variable in terms of another and then substitute this expression into other equations. Here鈥檚 what we did:
To solve a system, it often helps to express one variable in terms of another and then substitute this expression into other equations. Here鈥檚 what we did:
- In Equation (1): \( x^4 - y^3 = 17 \), we rearranged to solve for \( y^3 \), giving us: \( y^3 = x^4 - 17 \).
- This expression for \( y^3 \) is substituted into Equation (2) to eliminate \( y \) and find a solution in terms of \( x \).
Polynomial Equations
Polynomial equations are fundamental in algebra, involving terms like squares, cubes, and even higher powers of variables. Our problem introduces two polynomial equations, one quartic \( (x^4) \) and one cubic \( (y^3) \).
To solve these types of equations, you usually need to:
Handling polynomial equations is a critical skill in algebra, as variably driven setups often model real-world situations.
To solve these types of equations, you usually need to:
- Rearrange the equations to isolate higher power terms, such as \( x^4 \) or \( y^3 \).
- Factor or simplify where possible to reduce complexity.
Handling polynomial equations is a critical skill in algebra, as variably driven setups often model real-world situations.
Algebraic Manipulation
Algebraic manipulation involves rearranging and simplifying expressions to find solutions. This process allows you to make equations easier to handle and solve.
- In our solution, we manipulated the original equations into simpler forms. For example, we simplified \( 3x^4 + 5(x^4 - 17) = 53 \) into \( 8x^4 = 138 \) by distributing and combining like terms.
- We also turned division problems into simpler fractional equations. For example, we found \( x^4 = \frac{69}{4} \) by dividing both sides by 8.
