Solve a pair of equations using substitution
key notes:
To solve using substitution, follow these four steps:
Step 1: Isolate a variable.
Step 2: Plug the result of Step 1 into the other equation and solve for one variable.
Step 3: Plug the result of Step 2 into one of the original equations and solve for the other variable.
Step 4: State the solution.
The substitution method is easiest to use in one of these situations.
Learn with an example
Solve using substitution.
x = -9
-4x – 9y = -9
( _____ , _____ )
Use substitution to solve the simultaneous equations:
x = –9
–4x − 9y = –9
Step 1: Isolate a variable.
The variable x is already isolated in the first equation.
Step 2: Plug the result of Step 1 into the other equation and solve for one variable.
Plug x = –9 into the other equation, –4x − 9y = –9, and find the value of y.
–4x − 9y = –9
–4(–9) − 9y = –9 Plug in x = –9
36 − 9y = –9 Multiply
–9y = –45 Subtract 36 from both sides
y = 5 Divide both sides by –9
Step 3: Plug the result of Step 2 into one of the original equations and solve for the other variable.
The first equation already shows that x = –9, so it is not necessary to plug in and solve for x.
Step 4: State the solution.
Since x = –9 and y = 5, the solution is (–9, 5).
Solve using substitution.
y = 1
x – 7y = -3
( _____ , _____ )
Use substitution to solve the simultaneous equations:
y = 1
x − 7y = –3
Step 1: Isolate a variable.
The variable y is already isolated in the first equation.
Step 2: Plug the result of Step 1 into the other equation and solve for one variable.
Plug y = 1 into the other equation, x − 7y = –3, and find the value of x.
x − 7y = –3
x − 7(1) = –3 Plug in y = 1
x − 7 = –3 Multiply
x = 4 Add 7 to both sides
Step 3: Plug the result of Step 2 into one of the original equations and solve for the other variable.
The first equation already shows that y = 1, so it is not necessary to plug in and solve for y.
Step 4: State the solution.
Since x = 4 and y = 1, the solution is (4, 1).
Solve using substitution.
x = 10
-10x – 10y = -10
( ___ , ____ )
Use substitution to solve the simultaneous equations:
x = 10
–10x − 10y = –10
Step 1: Isolate a variable.
The variable x is already isolated in the first equation.
Step 2: Plug the result of Step 1 into the other equation and solve for one variable.
Plug x = 10 into the other equation, –10x − 10y = –10, and find the value of y.
–10x − 10y = –10
–10(10) − 10y = –10 Plug in x = 10
–100 − 10y = –10 Multiply
–10y = 90 Add 100 to both sides
y = –9 Divide both sides by –10
Step 3: Plug the result of Step 2 into one of the original equations and solve for the other variable.
The first equation already shows that x = 10, so it is not necessary to plug in and solve for x.
Step 4: State the solution.
Since x = 10 and y = –9, the solution is (10, –9).
lets practice:
