solve the system of two equations to get x1, x2 

Let's solve the system of equations using the elimination method:

\[ 
3x_1 + 5x_2 = 45 
\]

\[ 
8x_1 - 9x_2 = 56 
\]

We will use elimination to eliminate one of the variables. 

1. First, let's align the coefficients of \( x_1 \) by multiplying the first equation by 8 and the second equation by 3:

\[ 
8(3x_1 + 5x_2) = 8(45) \\
24x_1 + 40x_2 = 360 
\]

\[ 
3(8x_1 - 9x_2) = 3(56) \\
24x_1 - 27x_2 = 168 
\]

2. Next, we subtract the second equation from the first to eliminate \( x_1 \):

\[ 
(24x_1 + 40x_2) - (24x_1 - 27x_2) = 360 - 168 \\
24x_1 + 40x_2 - 24x_1 + 27x_2 = 192 \\
67x_2 = 192 \\
x_2 = \frac{192}{67} \\
x_2 \approx 2.8657 
\]

3. Substitute \( x_2 \) back into the first equation to find \( x_1 \):

\[ 
3x_1 + 5 \left( \frac{192}{67} \right) = 45 \\
3x_1 + \frac{960}{67} = 45 \\
3x_1 = 45 - \frac{960}{67} \\
3x_1 = 45 - 14.3284 \\
3x_1 = 30.6716 \\
x_1 = \frac{30.6716}{3} \\
x_1 \approx 10.2239 
\]

So, the solutions to the system are:

\[ 
x_1 \approx 10.2239 
\]

\[ 
x_2 \approx 2.8657 
\]

Would you like more details or have any questions?

Here are 5 questions you might find interesting:

1. Can you explain how to solve a system of equations using the substitution method?
2. How can I check if my solutions to a system of equations are correct?
3. What are some other methods to solve systems of linear equations?
4. How do you graphically represent the solution of a system of linear equations?
5. What are the conditions for a system of linear equations to have no solution, one solution, or infinitely many solutions?

**Tip:** When solving systems of equations, always simplify your answers to the maximum extent possible to ensure accuracy and clarity.

Learn how to solve the system of equations 3x1 + 5x2 = 45 and 8x1 - 9x2 = 56 using the elimination method. This problem involves solving simultaneous linear equations and demonstrates the application of algebraic methods to find precise solutions. Suitable for high school students.

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