Final Answer:
a. The solution to the system of linear equations x + y = 7 and 2x + y = 5 is x = -3 and y = 10.
b. The solution was obtained by solving the system of equations using algebraic methods. The variables x and y were determined by performing operations to isolate the variables in each equation. The resulting values of x = -3 and y = 10 satisfy both equations simultaneously, making them the solution to the given system of linear equations.
Step-by-step explanation:
a. To find the solution to the system of linear equations x + y = 7 and 2x + y = 5, algebraic methods such as substitution or elimination can be employed. In this case, subtracting the first equation from the second eliminates y, yielding x = -3. Substituting this value into the first equation then gives y = 10. Thus, the solution to the system is x = -3 and y = 10.
b. The solution was obtained through systematic manipulation of the equations. Subtracting the first equation from the second was chosen as the method for eliminating one variable. The resulting values of x = -3 and y = 10 were validated by substituting them back into both original equations to ensure they satisfy both equations simultaneously. This method demonstrates the fundamental principles of solving a system of linear equations through algebraic techniques.
Understanding the process of solving systems of linear equations is crucial in various fields, including mathematics, physics, and engineering. It allows for the determination of the values of multiple variables that satisfy a set of equations, providing solutions to real-world problems modeled by linear relationships.