Algebra - "Solve for x" Challenge¶
Problem Statement¶
Solve for \(x\) and \(y\). You should have coordinate pairs (\(x\), \(y\)) for each equation:
Step-by-Step Solution¶
To solve the equation:
we can follow these steps:
Step 1: Combine like terms¶
First, let's group the terms involving \(x^{4/3}\) on one side and \(x^{3/4}\) on the other side. We do this by subtracting \(6x^{4/3}\) and adding \(3x^{3/4}\) to both sides of the equation:
Combine like terms:
[ (2x^{4/3} - 6x^{4/3}) + (5x^{3/4} + 3x^{3/4}) = 0 ] [ -4x^{4/3} + 8x^{3/4} = 0 ]
Step 2: Isolate one of the terms¶
To simplify, isolate one of the terms. Here, isolate \(x^{4/3}\) or \(x^{3/4}\). Let's isolate \(x^{4/3}\):
Divide both sides by -4:
Step 3: Solve for \(x\)¶
To solve for \(x\), we need to eliminate the exponents. We can do this by dividing both sides by \(x^{3/4}\) to get:
Apply the Quotient Rule for Exponents which states \(\frac{a^m}{a^n} = a^{m-n}\):
First, find a common denominator to subtract the exponents:
Convert to a common denominator (which is 12):
So:
Step 4: Solve for \(x\)¶
Raise both sides to the power of the reciprocal of \(7/12\) to solve for \(x\):
Step 5: Simplify if necessary¶
This is already in its simplest form.
Solution:¶
The solution to the equation is:
Summary of Properties Used:¶
- Combining Like Terms: To simplify the equation by grouping similar terms.
- Isolating Variables: To simplify the equation to a form where the variable can be solved.
- Quotient Rule for Exponents: To combine terms with exponents and solve for \(x\).
- Exponentiation: To solve for \(x\) by raising both sides to the power of the reciprocal.