Substitution Method
How to solve a system of linear equations and inequalities by using their graphs: 3 examples and their solutions.
Examplex - y = 4, 2x + y = 5
Choose one of the equation.
Let's choose x - y = 4.
Change it to [x = ...] form.
Then x = y + 4.
You can also change it to [y = ...] form.
Put x = y + 4
into the other equation 2x + y = 5.
Then 2(y + 4) + y = 5.
Solve the equation 2(y + 4) + y = 5.
Then y = -1.
Put this y = -1
into x = y + 4.
Then x = (-1) + 4.
-1 + 4 = 3
So x = 3.
y = -1
x = 3
So
x = 3, y = -1
is the answer.
Examplex - y = 4, 2x - 2y = 8
Choose one of the equation.
Let's choose x - y = 4.
Change it to [x = ...] form.
Then x = y + 4.
Put x = y + 4
into the other equation 2x - 2y = 8.
Then 2(y + 4) - 2y = 8.
Solve the equation 2(y + 4) - 2y = 8.
Then the variable is removed
and you get 0 = 0.
0 = 0 is always true.
Just like this case,
when you get an equation
that is always true,
then the system has
infinitely many solutions.
So
infinitely many solutions
is the answer.
Examplex - y = 4, x - y = -3
Choose one of the equation.
Let's choose x - y = 4.
Change it to [x = ...] form.
Then x = y + 4.
Put x = y + 4
into the other equation x - y = -3.
Then (y + 4) - y = -3.
Solve the equation (y + 4) - y = -3.
Then the variable is removed
and you get 4 = -3.
4 = -3 is always false.
Just like this case,
when you get an equation
that is always false,
then the system has
no solution.
So
no solution
is the answer.