Answer:
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Explanation:
There are different methods to solve a system of equations, but two common ones are substitution and elimination:
Substitution method:
Solve one of the equations for one of the variables in terms of the other.
For example, from the first equation: 6x - 3y = 9 --> 2x - y = 3 --> y = 2x - 3.
Substitute the expression found for the variable into the other equation.
For example, in the second equation: 4x - 5y = -18 --> 4x - 5(2x - 3) = -18.
Solve for the remaining variable.
Simplifying the equation: 4x - 10x + 15 = -18 --> -6x = -33 --> x = 33/6 = 5.5.
Substitute the value found for the variable into one of the original equations to find the other variable.
For example, using the first equation: 6x - 3y = 9 --> 6(5.5) - 3y = 9 --> 33 - 3y = 9 --> y = (33-9)/3 = 8.
Therefore, the solution for the system of equations is x = 5.5 and y = 8, or (5.5, 8).
Elimination method:
Multiply one or both equations by constants that will make the coefficients of one of the variables equal in magnitude but opposite in sign.
For example, multiplying the first equation by 5 and the second equation by 2:
30x - 15y = 45
8x - 10y = -36.
Add or subtract the resulting equations to eliminate one of the variables.
Subtracting the second equation from the first: 30x - 8x - 15y + 10y = 45 - (-36) --> 22x - 5y = 81.
Solve for the remaining variable.
Solving for y: 22x - 5y = 81 --> 5y = 22x - 81 --> y = (22/5)x - 81/5.
Substitute the value found for the variable into one of the original equations to find the other variable.
Using the first equation: 6x - 3y = 9 --> 6x - 3[(22/5)x - 81/5] = 9 --> 6x - (66/5)x + 27 = 9 --> (24/5)x = -18 --> x = -15/2.
Substitute the values found for the variables into one of the original equations to check the solution.
Using the second equation: 4x - 5y = -18 --> 4(-15/2) - 5[(22/5)(-15/2) - 81/5] = -18, which simplifies to -18 = -18, confirming the solution.
Therefore, the solution for the system of equations is x = -15/2 and y = 33/5, or (-7.5, 6.6).