Answer :
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]
[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]
