I know there are methods out there but what do you think is the best or easiest method(s) of solving a system of equations.
Depends on the coefficients of the equation.
If in one of the equations I can solve for one of the variables without having to express it in fractions, I use substitution
3x + 7y = 9
4x + y = -3 ------> y = -4x-3, now sub that in the other equation
If not, I use elimination, that is, changing one or both of the equations to get a common coefficient for one of the variables, then subtracting or adding them to "eliminate" that variable.
2x+5y=6 ---times 3 -----> 6x+15y=18
3x-7y=1 ---times 2 -----> 6x-14y=2
now subtract them etc
The best or easiest method of solving a system of equations depends on the specific coefficients in the equations. There are two common methods: substitution and elimination.
The substitution method involves solving one equation for one variable and substituting that expression into the other equation. This works when one of the equations can be easily solved for a variable without having to express it in fractions. For example:
Equation 1: 3x + 7y = 9
Equation 2: 4x + y = -3
In this case, Equation 2 can be solved for y: y = -4x - 3. Then, you can substitute this expression for y into Equation 1, and solve for x. Once you have the value of x, you can substitute it back into either equation to solve for y.
The elimination method involves manipulating one or both of the equations to obtain a common coefficient for one of the variables, then adding or subtracting the equations to "eliminate" that variable. For example:
Equation 1: 2x + 5y = 6
Equation 2: 3x - 7y = 1
In this case, you can multiply Equation 1 by 3 and Equation 2 by 2 to obtain the same coefficient for x in both equations. Equation 1 becomes: 6x + 15y = 18, and Equation 2 becomes: 6x - 14y = 2. Now, you can subtract one equation from the other to eliminate x and solve for y. Substituting the found value of y back into either equation will allow you to solve for x.
Both methods have their advantages depending on the specific problem at hand. It's a good idea to try both methods and see which one leads to a simpler or more efficient solution for the particular system of equations you are working with.