Download Solving Linear Systems by Substitution

Algebra I
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SOL A.4 Solving Systems of Linear Equations by Substitution (Sec. 7.2)
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Solving Linear Systems by Substitution
Essential Quet
Essential Question(s): How do you solve a system of linear equations by substitution? When might
it be easier to solve a linear system by substitution instead of by graphing?
Steps to Solving a System of Linear Equations by Substitution:
1. Solve one equation for one of the variables.
(HINT: look to solve for the variable with a coefficient of 1).
2. Substitute the expression from Step 1 into the other equation & solve for the
other variable.
3. Substitute the value from Step 2 into the revised equation from Step 1 & solve.
4. Write your solution as an ordered pair.
5. Check your answer algebraically.
Example 1: Solve the following system of equations using substitution.
y = 3x + 2
x + 2y = 11
Equation 1
y = 3x + 2
Why do you substitute
the expression inside
parentheses?
Equation 2
x + 2y = 11
x + 2(_________) = 11
1: Solve one equation for x or y.
(Hint: Equation 1 is already solved for y!)
2: Substitute the expression from Step 1
into the other equation (Equation 2).
Solve for x.
3: Substitute _____ for _____ into
Equation 1 and solve for _________
The solution is _______________.
4. Write answer as an ordered pair.
Check:
5. Check solution in both equations.
y = 3x + 2
x + 2y = 11
Example 2: Solve the following system of equations using substitution.
4x – 7y = 10
y=x–7
Equation 1
4x – 7y = 10
Why change
4x – 7(x – 7) = 10
to
4x +(-7)(x – 7 ) = 10?
Equation 2
y=x–7
1: Solve one equation for x or y.
(Hint: Equation 2 is already solved for y!)
2: Substitute the expression from Step 1
into the other equation (Equation 1).
Solve for x.
4x – 7(_________) = 10
4x +(-7)(_________) = 10
3: Substitute _____ for _____ into
Equation 2 and solve for _________
4. Write answer as an ordered pair.
The solution is _______________.
5. Check solution in both equations.
Check:
4x – 7y = 10
y=x–7
Practice: Solve the following systems using substitution. Check your solutions.
1) y = 2x + 5
3x + y = 10
2)
x=y+3
2x – y = 5
Sometimes, you won’t be given an equation already solved for one variable.
In that case, you’ll need to solve one of the equations for one variable.
(Remember the HINT! – look to solve for the variable with a coefficient of 1)
Example 3: Solve the following system of equations using substitution.
x – 2y = -6
4x + 6y = 4
Equation 1
x – 2y = -6
Equation 2
4x + 6y = 4
1: Solve one equation for x or y.
Look for the variable with a coefficient of 1
2: Substitute the expression from Step 1
into the other equation.
Solve for the variable.
3: Substitute the value from Step 2 into the
revised equation from Step 1 and solve.
The solution is ________.
5. Check solution in both equations.
Check:
x – 2y = -6
Practice:
1) x – y = 3
x + 2y = -6
4. Write answer as an ordered pair.
4x + 6y = 4
2) 3x + y = -7
-2x + 4y = 0
Special Cases
Infinitely Many solutions: You are left with a __________ statement, regardless of x or y.
Example: y = -2x + 4
2x + y = 4
Equation 1
y = -2x + 4
Equation 2
2x + y = 4
No Solution: You are left with a ____________ statement, regardless of x or y.
Example: y = -2x
2x + y = 4
Equation 1
y = -2x
Summary:
Equation 2
2x + y = 4