91Ó°ÊÓ

The elimination method is a technique used to solve systems of linear equations by removing one variable. This makes it easier to solve for the remaining variable. Let's look at how this works with the given exercise. First, we have two equations:
1) \( x + y = 9 \)
2) \( 3x + 2y = 22 \)
Our goal is to eliminate one of the variables by adding or subtracting the equations.

First, multiply the first equation by 2 to align the coefficients of the variable we want to eliminate:
\( 2(x + y) = 2(9) \)
which simplifies to:
\( 2x + 2y = 18 \).

Now, subtract this new equation from the second equation:
\( (3x + 2y) - (2x + 2y) = 22 - 18 \).

This simplifies to:
\( x = 4 \).

This method efficiently helps you eliminate one variable to make solving for the other simpler.

The substitution method is another way to solve systems of linear equations, especially useful when one variable is already isolated. Here's how it works.

Start with the solution we found using elimination: \( x = 4 \). Now substitute \( x = 4 \) back into the first original equation:
\( x + y = 9 \)
which becomes:
\( 4 + y = 9 \).

Subtract 4 from both sides to solve for y:
\( y = 5 \).

In the substitution method, we replace one variable with an expression that contains the other variable, making it easier to find the solution.

Verifying the solution ensures that the values we found indeed satisfy the original equations. Let's verify our solution \( x = 4 \) and \( y = 5 \).

Substitute \( x = 4 \) and \( y = 5 \) back into both original equations:
First equation: \( x + y = 9 \) becomes:
\( 4 + 5 = 9 \)
which is correct.

Second equation: \( 3x + 2y = 22 \) becomes:
\( 3(4) + 2(5) = 12 + 10 = 22 \)
which is also correct.

By verifying, we confirm our solution is accurate and consistent with the original system of equations.