Final answer:
To solve each system of equations using the substitution method involves expressing one variable in terms of the other and then substituting this into the remaining equation. After solving for the first variable, substitute its value back into one of the original equations to find the corresponding value of the second variable.
Step-by-step explanation:
To solve the given systems of equations using the substitution method, we substitute one equation into another for one of the variables, then solve for the remaining variable.
First System
From the second equation: x − y = 3, we solve for x: x = y + 3.
Substitute x in the first equation: 6x + 2y = 82 with y + 3.
Now we have 6(y + 3) + 2y = 82.
Simplify and solve for y.
Substitute the value of y back into x = y + 3 to find x.
Second System
From the first equation: y = 4x − 1, we already have y expressed in terms of x.
Substitute y in the second equation: x + y = 9 with 4x − 1.
Now we have x + (4x − 1) = 9.
Simplify and solve for x.
Substitute the value of x back into y = 4x − 1 to find y.
Third System
From the second equation: y = 5 − x, we already have y expressed in terms of x.
Substitute y in the first equation: 4x + 3y = 17 with 5 − x.
Now we have 4x + 3(5 − x) = 17.
Simplify and solve for x.
Substitute the value of x back into y = 5 − x to find y.