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Substitution
WTK
Substitution – when you substitute the value of
one variable into an equation and solve for the
other variable.
Solving Systems by Substitution
• When solving by substitution, one equation
must be solved for a single variable.
– This mean, either x or y must be on one side of
the equal sign by itself.
• If your problem isn’t solved for a single
variable, you have to rewrite your equation.
• Once one equation is solved for a single
variable, substitute the equivalent expression
in for that variable in the other equation.
𝑦=6
Example #1:
2𝑥 + 𝑦 = 12
• 1st equation is solved for y
– We know 𝑦 = 6.
– When we see y in the second equation, plug in 6.
2𝑥 + 𝑦 = 12
2𝑥 + __________ = 12
Solution:
𝑥+𝑦 =5
Example #2:
𝑦 =3+𝑥
Step 1: Solve an equation
for one variable.
Step 2: Substitute.
Step 3: Solve the
equation.
Step 4: Plug back in to
find the other variable.
Solution:
The 2nd equation is
already solved for y.
𝑦 =3+𝑥
𝑥+𝑦 =5
𝑥 + ______________ = 5
𝑥 =3−𝑦
Example #3:
𝑥+𝑦 =7
Step 1: Solve an equation
for one variable.
Step 2: Substitute.
The 2nd equation is already
solved for x.
𝑥 =3−𝑦
𝑥+𝑦 =7
(_____________) + 𝑦 = 7
Step 3: Solve the equation.
Does 3 = 7?
Because this is not true, this
system has no solution.
2𝑥 + 𝑦 = 4
Example #4:
4𝑥 + 2𝑦 = 8
Step 1: Solve an equation
for one variable.
It is easiest to solve the 1st
equation for y.
2𝑥 + 𝑦 = 4
𝑦 = 4 − 2𝑥
2𝑥 + 𝑦 = 4
Example #4:
4𝑥 + 2𝑦 = 8
Step 2: Substitute.
𝑦 = 4 − 2𝑥
4𝑥 + 2(__________) = 8
Step 3: Solve the
equation.
Does 8 = 8?
Because this is true, this
system has infinite
solutions.
3𝑦 + 𝑥 = 7
Example #5:
4𝑥 − 2𝑦 = 0
Step 1: Solve an equation
for one variable.
It is easiest to solve the
2nd equation for x.
3𝑦 + 𝑥 = 7
𝑥 = 7 − 3𝑦
3𝑦 + 𝑥 = 7
Example #5:
4𝑥 − 2𝑦 = 0
Step 2: Substitute.
Step 3: Solve the
equation.
𝑥 = 7 − 3𝑦
4(__________) − 2𝑦 = 0
3𝑦 + 𝑥 = 7
Example #5:
4𝑥 − 2𝑦 = 0
Step 4: Plug back in to
find the other variable.
Solution:
𝑦=2
𝑥 = 7 − 3𝑦
𝑥 = 7 − 3(_______)