The two equations can be added directly to eliminate y because their y-coefficients are exact opposites (−4y and +4y). When equations with opposite coefficients for the same variable are added term by term, those terms cancel out to zero, leaving a one-variable equation in x.
Conditions: The system has been reduced to integer coefficients: 3x−4y=14 and 15x+4y=22.; The equations are aligned term-by-term before addition.
The two equations can be added directly to eliminate y because their y-coefficients are exact opposites (−4y and +4y). When equations with opposite coefficients for the same variable are added term by term, those terms cancel out to zero, leaving a one-variable equation in x.
Conditions: The system has been reduced to integer coefficients: 3x−4y=14 and 15x+4y=22.; The equations are aligned term-by-term before addition.
The method solves the congruences one by one by maintaining an arithmetic progression of candidate solutions. It starts with the sequence of numbers satisfying the first congruence, tests them against the second to find a common solution, and then updates the step size to the least common multiple (LCM) of the first two moduli.
Conditions: The unknown x is an integer.; The system consists of the congruences x ≡ 3 (mod 4), x ≡ 5 (mod 6), and x ≡ 2 (mod 5).
The method solves the congruences one by one by maintaining an arithmetic progression of candidate solutions. It starts with the sequence of numbers satisfying the first congruence, tests them against the second to find a common solution, and then updates the step size to the least common multiple (LCM) of the first two moduli.
Conditions: The unknown x is an integer.; The system consists of the congruences x ≡ 3 (mod 4), x ≡ 5 (mod 6), and x ≡ 2 (mod 5).