We are given the following system of equations:

1. \( 91 = 50i_1 + 15(i_1 - i_2) \)
2. \( -15 = 15(i_2 - i_1) \)

Let's solve these step by step.

### Step 1: Simplify both equations

For the first equation:
\[
91 = 50i_1 + 15(i_1 - i_2)
\]
Distribute the \( 15 \) on the right-hand side:
\[
91 = 50i_1 + 15i_1 - 15i_2
\]
Combine like terms:
\[
91 = 65i_1 - 15i_2
\]
This gives us:
\[
91 = 65i_1 - 15i_2 \quad \text{(Equation 1)}
\]

For the second equation:
\[
-15 = 15(i_2 - i_1)
\]
Distribute the \( 15 \):
\[
-15 = 15i_2 - 15i_1
\]
This simplifies to:
\[
-15 = 15i_2 - 15i_1 \quad \text{(Equation 2)}
\]

### Step 2: Solve the system of equations

Now we have two equations:
1. \( 91 = 65i_1 - 15i_2 \)
2. \( -15 = 15i_2 - 15i_1 \)

We can either solve by substitution or elimination. I'll use the elimination method here. Let's add both equations together:

\[
(91) + (-15) = (65i_1 - 15i_2) + (15i_2 - 15i_1)
\]
Simplifying both sides:
\[
76 = 50i_1
\]
Solving for \( i_1 \):
\[
i_1 = \frac{76}{50} = 1.52
\]

### Step 3: Substitute \( i_1 \) into one of the original equations

Substitute \( i_1 = 1.52 \) into Equation 2:
\[
-15 = 15i_2 - 15(1.52)
\]
Simplify:
\[
-15 = 15i_2 - 22.8
\]
Add \( 22.8 \) to both sides:
\[
7.8 = 15i_2
\]
Solve for \( i_2 \):
\[
i_2 = \frac{7.8}{15} = 0.52
\]

### Final Solution:
\[
i_1 = 1.52, \quad i_2 = 0.52
\]

Would you like further details or have any questions?

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Here are 5 related questions to expand your understanding:

1. How do you solve a system of equations using substitution?
2. What is the difference between substitution and elimination methods in solving equations?
3. How can you check the solution of a system of linear equations?
4. What are some practical applications of systems of equations in physics or engineering?
5. How can matrix methods like Gaussian elimination be used to solve larger systems of equations?

**Tip:** Always double-check your solution by substituting the values back into the original equations to verify the results!

91 = 50i1 + 15(i1 - i2), -15 = 15(i2 - i1)

Learn how to solve the linear system of equations 91 = 50i1 + 15(i1 - i2) and -15 = 15(i2 - i1) using the elimination method. This step-by-step solution covers key algebraic concepts like simplifying equations and solving for variables. Suitable for students in Grades 10-12.

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