Download Solving Systems with Substitution

Dr. Fowler  CCM
Solving Systems of Equations
By Substitution – Easier
Solving a system of equations by substitution
Step 1: Solve an equation
for one variable.
Pick the easier equation. The goal
is to get y= ; x= ; a= ; etc.
Step 2: Substitute
Put the equation solved in Step 1
into the other equation.
Step 3: Solve the equation.
Get the variable by itself.
Step 4: Plug back in to find
the other variable.
Substitute the value of the variable
into the equation.
Step 5: Check your
solution.
Substitute your ordered pair into
BOTH equations.
EXAMPLE 1
Solve by substitution:
2 x  7 y  12
x  2 y
The second is solved for X. Substitute
this into OTHER equation for X:
2  2 y   7 y  12
4 y  7 y  12
3 y 1 2

3
3
y  4
Substitute found y
into other equation:
x  2 y
x  2  4
x 8
8, 4
The solution set found by the substitution method will be the same as
the solution found by graphing. The solution set is the same; only the
method is different. ALWAYS put answer in Alphabetical order. (x,y)
2) Solve the system using substitution
x+y=5
y=3+x
Step 1: Solve an equation
for one variable.
Step 2: Substitute
Step 3: Solve the equation.
The second equation is
already solved for y!
x+y=5
x + (3 + x) = 5
2x + 3 = 5
2x = 2
x=1
2) Solve the system using substitution
x+y=5
y=3+x
Step 4: Plug back in to find
the other variable.
Step 5: Check your
solution.
x+y=5
(1) + y = 5
y=4
(1, 4)
(1) + (4) = 5
(4) = 3 + (1)
The solution is (1, 4). What do you think the answer
would be if you graphed the two equations?
3) Solve the system using substitution
x=3–y
x+y=7
Step 1: Solve an equation
for one variable.
Step 2: Substitute
Step 3: Solve the equation.
The first equation is
already solved for x!
x+y=7
(3 – y) + y = 7
3=7
The variables were eliminated!!
This is a special case.
Does 3 = 7? FALSE!
When the result is FALSE, the answer is NO SOLUTIONS.
4) Solve the system using substitution
2x + y = 4
4x + 2y = 8
Step 1: Solve an equation
for one variable.
Step 2: Substitute
Step 3: Solve the equation.
The first equation is
easiest to solved for y!
y = -2x + 4
4x + 2y = 8
4x + 2(-2x + 4) = 8
4x – 4x + 8 = 8
8=8
This is also a special case.
Does 8 = 8? TRUE!
When the result is TRUE, the answer is INFINITELY MANY SOLUTIONS.
Example 5)
Solve the following system of equations
using the substitution method.
y = 3x – 4 and 6x – 2y = 4
The first equation is already solved for y. Substitute
this into second equation.
6x – 2y = 4
6x – 2(3x – 4) = 4
6x – 6x + 8 = 4
8=4
(substitute)
(use distributive property)
(simplify the left side)
Does 8=4? FALSE.
Examples like this – the answer is NO SOLUTION Ø.
If you graphed them, they would be PARALLEL LINES.
EXAMPLE 6
Solve the system by the substitution method.
2 x  7 y  12
x  3 2y
The second is solved for X. Substitute
this into OTHER equation for X:
2  3  2 y   7 y  12
6  4 y  7 y  12
6  3 y  6  12  6
3 y 1 8

3
3
y  6
Substitute found y
into other equation:
x  3  2  6
x  3 12
x  15
15, 6
Example #7:
y = 4x
3x + y = -21
Step 1: Solve one equation for one variable.
y = 4x
(This equation is already solved for y.)
Step 2: Substitute the expression from step one into
the other equation.
3x + y = -21
3x + 4x = -21
Step 3: Simplify and solve the equation.
7x = -21
x = -3
y = 4x
3x + y = -21
Step 4: We found x = -3. Now, substitute this into
either original equation to find y:
y = 4x (easiest)
y = 4(-3)
y = -12
Solution to the system is (-3, -12).
Excellent Job !!!
Well Done
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