If 2x + 3y = 16 and 2x - 3y = 36, the value of x is:

To solve a system of two linear equations in two variables, the elimination method removes one variable by adding or subtracting the equations when that variable's coefficients are equal in magnitude but opposite (or equal) in sign, leaving a single-variable equation to solve directly.

1. Write the two equations: 2x + 3y = 16 ...(1) and 2x - 3y = 36 ...(2).

2. The y-coefficients (+3 and -3) are exact opposites, so adding equation (1) and equation (2) eliminates y.

3. Adding: (2x + 3y) + (2x - 3y) = 16 + 36, which gives 4x = 52.

4. Divide both sides by 4: x = 13.

Cross-check: substituting x = 13 into equation (1) gives 2(13) + 3y = 16, so y = -10/3. Substituting x = 13 and y = -10/3 into equation (2) gives 2(13) - 3(-10/3) = 26 + 10 = 36, which matches equation (2) exactly, confirming the value of x.