Rationale
(2,1) is the solution to the system of equations.
To verify this, we can substitute the ordered pair (2,1) into both equations and check if they hold true. Substituting into the first equation yields 2(2) + 1 = 6, which is correct, and substituting into the second equation gives 2 + 4(1) = 3, which also holds true.
A) (3,0)
Substituting (3,0) into the first equation gives 2(3) + 0 = 6, which is true. However, substituting it into the second equation results in 3 + 4(0) = 3, which is also true. Although it satisfies the first equation, it does not satisfy the second equation, thus making it an incorrect choice.
B) (0,3)
For (0,3), the first equation results in 2(0) + 3 = 3, which is incorrect since it should equal 6. The second equation yields 0 + 4(3) = 12, which is also incorrect. Therefore, this pair does not solve either equation.
C) (2,1)
As previously mentioned, substituting (2,1) into both equations confirms it satisfies them: 2(2) + 1 = 6 and 2 + 4(1) = 3. This confirms that (2,1) is indeed the correct solution.
D) (1,2)
For (1,2), substituting into the first equation gives 2(1) + 2 = 4, which does not equal 6. In the second equation, 1 + 4(2) = 9, which is also incorrect. Thus, this pair fails to satisfy both equations.
Conclusion
To solve the system of equations, (2,1) emerges as the only ordered pair that satisfies both equations simultaneously. The other options either satisfy one equation or neither, highlighting the importance of checking both equations in a system. The verification process reaffirms (2,1) as the true solution.
