The cost to join an art museum
is P600. If you are a member,
you can take a lesson at the
museum for P20 each. If you
are not a member, lessons cost
P60 each. Write an equation to
find the number of x of lessons
after which the total cost y of
the lessons with membership is
the same as the total cost of
lessons without a membership.
Getting Ready!
1. Complete the table below
for x + y = 4.
X
0
1
2
3
4
5
y
2. Complete the table below
for 2x – y = 5.
X
y
0
1
2
3
4
5
Can you
name a pair
that satisfies
both
equation?
Linear system
Consist of two or more linear equations in the same variables
x + 2y = 7 and 3x – 2y = 5
Is an ordered pair that satisfies each of the equation in the
systems
X
0
1
2
3
4
5
y
Point of
intersection
X
y
0
1
2
3
4
5
1) -5x + y = 0 and 5x + y = 10
y
=
-5x
+
10
y = 5x
5
=
-5(1)
+
10
5 = 5(1)
5
=
-5
+10
5=5
5=5
2) x – y = 5 and 3x + y = 3
y = x – 5 y = -3x + 3
-3 = 2 – 5 -3 = -3(2) + 3
-3 = -6 + 3
-3 = -3
-3 = -3
It has at least
one solution.
Steps in Solving for Systems
by Graphing
3) x + y = 4 and 2x – y = 5
y = -x + 4
1 = -(3) + 4
1=1
y = 2x - 5
1 = 2(3) – 5
1=6-5
1=1
4) x – y = 1 and x + y = 3
y=x–1
1=2–1
1=1
y = -x + 3
1 = -(2) + 3
1 = -2 + 3
1=1
Homework
Solve the following linear systems
by graphing.
1. y = -x + 3 and y = x + 1
2. 3x + y = 15 and y = -15
1. y = -x + 3
y=x+1
2. 3x + y = 15
y = -15
A gardening company
placed orders with a nursery.
One was for 13 bushes
and 4trees, and totaled P487.
The second order was
for 6 bushes and 2 trees, and
totaled P232. The bill doesn't
tell the amount of per item.
What were the costs of one bush
and of one tree?
Getting Ready!
Simplify the following:
1. 3(2x + 3)
2. -4(3x – 4)
Substitute x – 3 for y and
simplify the following:
1. 5y
2. 3y + 2
3. 2(y+3)
Solve for x and y
1. y = 2x + 1 and 3x + 2y = 9
Can you solve
the system of
linear equation
by
substitution?
y=2x + 1 and 3x + 2y = 9
Eq. 1 is
already
solved for y. 3x + 2y = 9
eq. 2
3x + 2(2x + 1) = 9
3x + 4x + 2 = 9
7x = 9-2
7x = 7
x=1
y = 2x + 1
eq. 1
y = 2(1) + 1
y=2+1
y=3
a + b = 7 and 3a + 2b = 16
a+b=7
b = -a + 7
eq. 1
b = -a + 7
b = -(2) + 7
b = -2 + 7
b=5
eq. 1
3a + 2b = 16
eq. 2
3a + 2(-a + 7) = 16
3a – 2a + 14 = 16
a = 16 - 14
a=2
Steps in Solving for Systems
by Substitution
x – 2y = -6 and 4x + 6y = 4
x – 2y = -6
x = 2y - 6
x = 2y - 6
x = 2(2) - 6
x=4-6
x = -2
4x + 6y = 4
eq. 2
4(2y – 6) + 6y = 4
8y – 24 + 6y = 4
14y = 4 + 24
14y = 28
y =2
eq. 1
m = 2n + 5 and 3n + m = 10
Eq. 1 is
already solved
for m.
3n + m = 10
eq. 2
3n + (2n + 5) = 10
3n + 2n + 5 = 10
5n = 10 - 5
5n = 5
m = 2n + 5
m = 2(1) + 5
m=2+5
m=7
eq. 1
Homework
Solve the following linear systems by
substitution
1. x = y + 3 and 2x – y = 5
2. 11a – 7b = -14 and a- 2b =-4
Seatwork
Solve for the following:
(3x -2y) + (3x + 3y)
=
(x + y)
- (x - 4y)
(-2x - 4y)
+ (-4x + 4y)
(3x - 4y) + (-3x + 2y)
=
Developing Skills
Add the following equation.
y
x + 2y = 5
3x – 2y = -1
x + 2y = 5
1 + 2y = 5
2y = 5-1
2y = 4
y=2
Which
Solve
for the
variable
is
remaining
eliminated?
variable.
Add the following equation.
2a – 3b = 14
a + 3b = -2
b
a + 3b = -2
4 + 3b = -2
3b = -2 - 4
3b = -6
b = -2
Solve for the
Which
remaining
variable
variable.is
eliminated?
Developing Skills
Add the following equation.
x
2x + 3y = 11
2x - 5y = 13
2x + 3y = 11
2x + 3(3) = 11
2x = 11 - 9
2x = 2
x=1
Which
Solve for the
variable is
remaining
eliminated?
variable.
Steps in Solving for Systems
by adding and subtracting
Add the following equation.
2a – 3b = 26
-2a - 3b = -2
b
2a - 3b = 26
2a – 3(-4) = 26
2a = 26 - 12
2a = 24
a = 12
Solve for the
Which
remaining
variable
variable.is
eliminated?
5x + 2y = 16
3x – 4y = 20 (5x + 2y = 16)2
10x + 4y = 32
3x – 4y = 20
Can we eliminate
a variable by
adding and
subtracting?
10x + 4y = 32
3x – 4y = 20
3x – 4y
3(4) – 4y
– 4y
– 4y
y
=
=
=
=
=
20
20
20 -12
8
-2
6x + 5y = 19
(2x + 3y = 5 )-3
6x + 5y = 19
-6x - 9y = -15
6x + 5y = 19
6x +5(-1) = 19
6x = 19 + 5
6x = 24
x = 4
( 4x + 5y = 35 ) 2
(-3x + 2y = -9 ) 5
8x + 10y = 70
-15x+10y = -45
4x + 5y = 35
4(5) +5y = 35
5y = 35 - 20
5y = 15
y = 3
Seatwork
Homework
Solve the following linear systems by
Elimination
1. 3x - 7y = 5 and 9y= 5x + 5
2. 3a + 2b = 4 and 2b =8 – 5a
Short Quiz
Solve the following linear systems by
Elimination
1. x + 4y = 22 and 4x – y = 3
2. 2x – 3y = 10 and x + 3y = -8
3.3x – y = 5 and 5x + 2y = 23
It has at least
one solution.
Inconsistent Linear System
> A linear system that has no solution.
The slopes of
an
inconsistent
linear system
is equal.
Dependent Linear System
A linear system that has infinitely many solution.
It has identical graph
The slopes and
y-intercept is
equal
Number of
solutions
Slopes and
y - intercept
One solution
Different slopes
No solution
Same Slope
Different y-intercept
Infinitely many Same Slope
solutions
Same y-intercept
Bell Work
Determine whether the statement is true or
false.
1. A solution of a linear system is an
ordered pair (x,y)
2. Graphically, the solution of an
independent system is the point of
intersection.
3. An independent system of equation
has no solution.
4. The graph of an inconsistent system is
identical.
5. A system of linear equation can either
have one solution or no solution.
II. Answer the following.
6. Is (2,3) a solution of the system
3x + 4y = 18
2x – y = 1
?
7. Is ( 1, -2 ) a solution of the system
3x – y = 14
2x + 5y = 8 ?
8. Is (-1,3) a solution of the system
4x – y = -5
2x + 5y = 13 ?
9. Is (0,0) a solution of the system
4x + 3y = 0
2x – y = 1 ?
10. Is (2,-3) a solution of the system
y = 2x – 7
3x – y = 9
?
Seatwork:
Without graphing, determine whether
the linear system is independent,
inconsistent, or dependent.
1. y = -9x + 5
2. 3x + y = 6
y = 4x – 8
3x + y = -8
3. x + y = 3
2x + 2y = 6
4. x – y = 3
x+y=5
Seatwork:
Without graphing, determine whether
the linear system is independent,
inconsistent, or dependent.
1. y = -9x + 5
2. 3x + y = 6
y = 4x – 8
3x + y = -8
3. x + y = 3
2x + 2y = 6
4. x – y = 3
x+y=5
5. 3x – y = 3
2x + y = 2
6. 2x – y = 4
x+y=5
7. x + 2y = 6
x – 2y = 3
8. 5x – 2y = 10
3x + 2y = 6
9. 4x – 5y = 20
8x – 10y = 12
10. 8x + 2y = 6
y + 4x = -1
Seatwork
Homework
Solve the following linear systems and
indentify the type of system.
1. y = 7x + 13 and -21x + 3y = 39
2. x – 2y = 7 and –x + 2y = 7
System of Linear Inequalities
• Consist of two or more linear inequalities in
the same variable.
Solution of a System of Linear
Inequalities
• Is an ordered pair that is a solution for both
linear system.
Graph of a System of Linear
Inequalities
• Is the graph of all solutions of a system.
y > -x – 2 and y ≤ 3x + 6
y > -x – 2 and y ≤ 3x + 6
x > 1 and x <4
x > 1 and x <4
y > -3 and y < 4
y > -3 and y < 4
Steps in Graphing a system of Linear
Inequalities
Step 1: Graph each inequality.
Step 2: Find the intersection of the shaded part.
The graph of a system is its intersection.
y ≥ -1, x > -2 and x + 2y ≤ 4
y ≥ -1, x > -2 and x + 2y ≤ 4
Seatwork
Homework
Solve the following linear inequalities.
1. y ≤ x – 3 and y > -2x - 1
2. x < 8 and x > -2
y ≤ x – 3 and y > -2x - 1
X < 2 and x > -2
Problem Solving
1. There are 20 animals in a pen compose of dog and duck. If the
number of their legs is 52, how many dogs are there? How many
chickens are there?
2. A gardening company placed orders with nursery. One was fo