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In a system of equations, an equivalent system can be created by replacing one equation with the sum of two or more equations in the system or by replacing an equation with a multiple of itself. An equation can be also replaced by the sum of that equation and a multiple of another equation in the system.
The system described above is shown below. ax+by=c dx+ey+α (ax+by)=f+α c It is already known that ( x_1, y_1) is a solution to the first equation of this new system. It needs to be verified that the ordered pair is also a solution to the second equation. a x_1+b y_1=c ✓ d x_1+e y_1+α (a x_1+b y_1)? =f+α c Since ( x_1, y_1) is a solution to the original system, it is known that a x_1+b y_1=c and that d x_1+e y_1=f. These two expressions can be substituted into the second equation above.
Therefore, ( x_1, y_1) is also a solution to the second equation of the new system. This means that this ordered pair is a solution to the system formed by the first equation of the original system and the sum of the first equation and a multiple of the second equation. a x_1+b y_1=c ✓ d x_1+e y_1+α (a x_1+b y_1)=f+α c ✓