To solve the system of equations using the substitution method, we first need to express one variable in terms of the other from one of the equations, and then substitute it into the other equation.
We start with the two equations:
- \( 2x + 2y = 6 \)
- \( 3x + 2y = 9 \)
Step 1: Solve one of the equations for one of the variables
We'll start with the first equation: \[ 2x + 2y = 6 \]
Divide everything by 2 to simplify: \[ x + y = 3 \]
Now, we can solve for \( y \): \[ y = 3 - x \]
Step 2: Substitute this expression into the second equation
Now, we substitute \( y = 3 - x \) into the second equation: \[ 3x + 2y = 9 \] Replace \( y \): \[ 3x + 2(3 - x) = 9 \]
Distribute the 2: \[ 3x + 6 - 2x = 9 \]
Combine like terms: \[ x + 6 = 9 \]
Step 3: Solve for \( x \)
Subtract 6 from both sides: \[ x = 3 \]
Step 4: Substitute back to find \( y \)
Now that we have \( x \), we substitute it back into the equation we derived for \( y \): \[ y = 3 - x = 3 - 3 = 0 \]
Conclusion
The solution to the system of equations is: \[ x = 3, \quad y = 0 \]
Thus, the solution is: \[ (3, 0) \]
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