Answer:
To determine the best method for solving the given system of equations, let's analyze both substitution and elimination methods.
1. Substitution Method:
The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This allows us to solve for the remaining variable. The steps involved in using the substitution method are as follows:
Step 1: Solve one equation for one variable.
In this case, we can solve the second equation, y = 3x - 3, for y.
Step 2: Substitute the expression found in step 1 into the other equation.
Substituting y = 3x - 3 into the first equation, we get:
2x + 5(3x - 3) = 19
Step 3: Simplify and solve for x.
Expanding and simplifying the equation gives us:
2x + 15x - 15 = 19
17x - 15 = 19
17x = 34
x = 2
Step 4: Substitute the value of x back into one of the original equations to solve for y.
Using the second equation, we substitute x = 2:
y = 3(2) - 3
y = 6 - 3
y = 3
Therefore, the solution to the system of equations is x = 2 and y = 3.
2. Elimination Method:
The elimination method involves adding or subtracting equations to eliminate one variable. The steps involved in using the elimination method are as follows:
Step 1: Multiply one or both equations by suitable constants to make the coefficients of one variable equal or opposite.
In this case, we can multiply the second equation by 5 to make the coefficients of y equal in magnitude but opposite in sign.
Step 2: Add or subtract the equations to eliminate one variable.
Adding the modified equations, we get:
2x + 5y = 19
15x - 15y = -15
Combining like terms, we have:
17x - 10y = 4
Step 3: Solve the resulting equation for one variable.
In this case, we can solve for x:
17x - 10y = 4
17x = 10y + 4
x = (10y + 4)/17
Step 4: Substitute the value of x back into one of the original equations to solve for y.
Using the first equation, we substitute x = (10y + 4)/17:
2((10y + 4)/17) + 5y = 19
Simplifying and solving for y, we get: