Solve the system of equations: 3*2^x - 4*7^y = -172 and 7*2^x + 2*7^y = 154.

Steps to Solve

1. Label the Equations

Let's label the two equations for easier reference:

• Equation (1): ( 3 \cdot 2^x - 4 \cdot 7^y = -172 )
• Equation (2): ( 7 \cdot 2^x + 2 \cdot 7^y = 154 )

2. Rearranging Equation (1)

From Equation (1), isolate one of the variables. We can express ( 4 \cdot 7^y ) in terms of ( 2^x ): [ 4 \cdot 7^y = 3 \cdot 2^x + 172 ] Thus, [ 7^y = \frac{3 \cdot 2^x + 172}{4} ]

3. Substituting in Equation (2)

Now, substitute the expression for ( 7^y ) into Equation (2): [ 7 \cdot 2^x + 2 \cdot \frac{3 \cdot 2^x + 172}{4} = 154 ]

4. Simplifying the Equation

Multiply through by 4 to eliminate the fraction: [ 28 \cdot 2^x + 2(3 \cdot 2^x + 172) = 616 ] This simplifies to: [ 28 \cdot 2^x + 6 \cdot 2^x + 344 = 616 ]

5. Combining Like Terms

Now combine the ( 2^x ) terms: [ 34 \cdot 2^x + 344 = 616 ]

6. Isolate ( 2^x )

Subtract 344 from both sides: [ 34 \cdot 2^x = 272 ] Now divide by 34: [ 2^x = 8 ] Taking the logarithm or recognizing powers of 2 gives: [ x = 3 ]

7. Finding ( y )

Substitute ( x = 3 ) back into the expression for ( 7^y ): [ 7^y = \frac{3 \cdot 2^3 + 172}{4} = \frac{24 + 172}{4} = \frac{196}{4} = 49 ] Taking roots, we find: [ 7^y = 7^2 \implies y = 2 ]