Sure, let's solve the given system of equations step-by-step. The equations are:
[tex]\[
\begin{cases}
3(2^x) + 3y - 2 = 25 \\
2^x - 3y + 1 = -19
\end{cases}
\][/tex]
### Step 1: Simplify the equations
First, we'll simplify both equations by isolating the constants.
Equation 1:
[tex]\[
3(2^x) + 3y - 2 = 25
\][/tex]
Add 2 to both sides:
[tex]\[
3(2^x) + 3y = 27
\][/tex]
Divide everything by 3:
[tex]\[
2^x + y = 9
\][/tex]
This is our simplified first equation.
Equation 2:
[tex]\[
2^x - 3y + 1 = -19
\][/tex]
Subtract 1 from both sides:
[tex]\[
2^x - 3y = -20
\][/tex]
### Step 2: Solve for [tex]\(2^x\)[/tex]
Now we have the system:
[tex]\[
\begin{cases}
2^x + y = 9 \\
2^x - 3y = -20
\end{cases}
\][/tex]
Let's denote [tex]\(2^x\)[/tex] by [tex]\(a\)[/tex] for convenience:
[tex]\[
\begin{cases}
a + y = 9 \\
a - 3y = -20
\end{cases}
\][/tex]
### Step 3: Solve the first equation for [tex]\(y\)[/tex]
From the first equation:
[tex]\[
a + y = 9
\][/tex]
Solving for [tex]\(y\)[/tex]:
[tex]\[
y = 9 - a
\][/tex]
### Step 4: Substitute [tex]\(y\)[/tex] into the second equation
Substitute [tex]\(y = 9 - a\)[/tex] into the second equation [tex]\(a - 3y = -20\)[/tex]:
[tex]\[
a - 3(9 - a) = -20
\][/tex]
Distribute and simplify:
[tex]\[
a - 27 + 3a = -20
\][/tex]
Combine like terms:
[tex]\[
4a - 27 = -20
\][/tex]
Add 27 to both sides:
[tex]\[
4a = 7
\][/tex]
Divide by 4:
[tex]\[
a = \frac{7}{4}
\][/tex]
### Step 5: Substitute [tex]\(a\)[/tex] back to find [tex]\(x\)[/tex]
Recall that [tex]\(a = 2^x\)[/tex]:
[tex]\[
2^x = \frac{7}{4}
\][/tex]
By taking the logarithm base 2 of both sides to solve for [tex]\(x\)[/tex]:
[tex]\[
x = \log_2\left(\frac{7}{4}\right)
\][/tex]
### Step 6: Find the value of [tex]\(y\)[/tex]
We already have [tex]\(y = 9 - a\)[/tex]:
[tex]\[
y = 9 - \frac{7}{4}
\][/tex]
Convert 9 to a fraction with a denominator of 4:
[tex]\[
9 = \frac{36}{4}
\][/tex]
Now subtract:
[tex]\[
y = \frac{36}{4} - \frac{7}{4} = \frac{29}{4}
\][/tex]
### Conclusion
So, the solutions to the system of equations are:
[tex]\[
x = \log_2\left(\frac{7}{4}\right)
\][/tex]
and
[tex]\[
y = \frac{29}{4}
\][/tex]
The simplified form can be written as:
[tex]\[
(x, y) \approx \left(-2 + \log_2(7), \frac{29}{4}\right)
\][/tex]
