Download Solving Systems with Substitution

Warm Up
Solve each equation for x.
2. y = 3x – 4
1. y = x + 3
x=y–3
Simplify each expression.
3. 2(x – 5)
2x – 10
4. 12 – 3(x + 1)
9 – 3x
2
Evaluate
each expression for the given value of x.
3
5. x + 8 for x = 6
6. 3(x – 7) for x = 10
12
9
Lesson 6.2 Solving Systems
by Substitution
Objective
California
Standards
9.0 Students solve a system of two
linear equations in two variables
algebraically and are able to interpret the
answer graphically. Student are able to solve a
system of two linear inequalities in two
variables and to sketch the solution sets.
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: substitute 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.
1) Solve the system using substitution
x+y=5
y=3+x
Step 1: Solve an equation
for one variable.
Step 2: Substitute
The second equation is
already solved for y!
x+y=5
x+(
)=5
2x + 3 = 5
Step 3: Solve the equation.
Now you now the x-value is 1.
Now lets find the value for y
-3 -3
2x = 2
x=1
1) Finding the y-value
x+y=5
y=3+x
Step 4: substitute 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?
Which answer checks correctly?
3x – y = 4
x = 4y - 17
1.
2.
3.
4.
(2, 2)
(5, 3)
(3, 5)
(3, -5)
2) Solve the system using substitution
3y + x = 7
4x – 2y = 0
Step 1: Solve an equation
for one variable.
Step 2: Substitute
It is easiest to solve the
first equation for x.
3y + x = 7
-3y
-3y
x = -3y + 7
4x – 2y = 0
4(
) – 2y = 0
2) Solve the system using substitution
3y + x = 7
4x – 2y = 0
Step 3: Solve the equation.
Step 4: Plug back in to find
the other variable.
4( -3x+ 7) – 2y = 0
-12y + 28 – 2y = 0
-14y + 28 = 0
-14y = -28
y=2
4x – 2y = 0
4x – 2( ) = 0
4x – 4 = 0
4x = 4
x=1
2) Solve the system using substitution
3y + x = 7
4x – 2y = 0
Step 5: Check your
solution.
(1, 2)
3(2) + (1) = 7
4(1) – 2(2) = 0
When is solving systems by substitution easier
to do than graphing?
When only one of the equations has a variable
already isolated (like in example #1).
If you solved the first equation for x, what
would be substituted into the bottom equation.
2x + 4y = 4
3x + 2y = 22
1.
2.
3.
4.
-4y + 4
-2y + 2
-2x + 4
-2y+ 22
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
)+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 (
)=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.
What does it mean if the result is “TRUE”?
1.
2.
3.
The lines intersect
The lines are parallel
The lines are coinciding
Application
The sum of Maria’s sister and brother ages is 18. Her
brother is four years older than her sisters. Find the ages of
Maria’s sister and her brother.
Let x = her brother’s age
Let y = her sister’s age
X=4+7
X = 11
Brother’s age = 11
Sister’s age = 7
x + y = 18
x=4+y
4 + y + y = 18
4 + 2y = 18
-4
= -4
2y= 14
Y=7
Lesson Quiz: Part I
Solve each system by substitution.
y = 2x
(–2, –4)
1.
x = 6y – 11
2.
3x – 2y = –1
3.
–3x + y = –1
x–y=4
(1, 2)
Lesson Quiz: Part II
4.
Plumber A charges $60 an hour. Plumber B
charges $40 to visit your home plus $55 for
each hour. For how many hours will the total
cost for each plumber be the same? How
much will that cost be? If a customer thinks
they will need a plumber for 5 hours, which
plumber should the customer hire? Explain.
8 hours; $480; plumber A: plumber
A is cheaper for less than 8 hours.