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Solving Equations with
Variables on Both Sides
Students will be able to solve equations
with variables on both sides and solve
equations containing grouping symbols.
Warm-Up #11 (3/6/2017)
1.
2.
3.
Is (3, 4) a solution to the equation
Is (-1, 2) a solution to the equation
Solve for x. 10𝑥 + 5 = 2𝑥 − 15.
𝑥−𝑦 =8
𝑥 + 3𝑦 = 5
Homework (3/6/2017)
Worksheet: Equations with Infinite
and No Solutions (ODD #, front and
back)
You need to get the variables on
one side of the equation. It does
not matter which variable you
move. Try to move the one that
will keep your variable positive.
1.
2.
3.
4.
5.
6.
1) Solve 3x + 2 = 4x - 1
Draw “the river”
- 3x
- 3x
Subtract 3x from
2 = x-1
both sides
Simplify
+
1
+
1
Add 1 to both
sides
3
=
x
Simplify
Check your
3(3) + 2 = 4(3) - 1
answer
9 + 2 = 12 - 1
1.
2.
3.
4.
5.
6.
7.
8.
2) Solve 8y - 9 = -3y + 2
+ 3y
+ 3y
Draw “the river”
11y – 9 =
2
Add 3y to both sides
Simplify
+9
+9
Add 9 to both sides
Simplify
11y = 11
Divide both sides by
11
11
11
Simplify
Check your answer
y=1
8(1) - 9 = -3(1) + 2
What is the value of x if
3 - 4x = 18 + x?
1. -3
2.
3.
1
3
1
3
4. 3
Answer Now
1.
3) Solve 4 = 7x - 3x
Draw “the river”
4 = 4x
– Notice the variables
4 4
are on the same side!
Combine like terms
1=x
Divide both sides by 4
2.
3.
4. Simplify
5. Check your answer
4 = 7(1) - 3(1)
1.
2.
3.
4.
5.
6.
7.
4) Solve -7(x - 3) = -7
Draw “the river”
-7x + 21 = -7
Distribute
- 21 - 21
Subtract 21 from both
sides
-7x
= -28
Simplify
Divide both sides by -7
-7
-7
Simplify
Check your answer
x=4
-7(4 - 3) = -7
-7(1) = -7
What is the value of x if
3(x + 4) = 2(x - 1)?
1.
2.
3.
4.
-14
-13
13
14
Answer Now
5) Solve
3 1
1
3
 x  x
8 4
2
4
1. Draw “the river”
3
1
1
3
2. Clear the fraction – (8)  (8) x  (8) x  (8)
8
4
2
4
multiply each term by
the LCD
3 - 2x = 4x – 6
3. Simplify
+ 2x +2x
4. Add 2x to both sides
3
= 6x – 6
5. Simplify
+6
+6
6. Add 6 to both sides
7. Simplify
9
= 6x
8. Divide both sides by 6
6
6
3
9. Simplify
or 1.5 = x
10. Check your answer
2
3 1
1
3
 1.5   1.5  
8
4
2
4
A system of linear equations is a set of
two or more linear equations in the
same variable. An example is shown
below:
y= 𝑥 + 1
y= 2𝑥 − 7
A solution to a system of equations is an ordered pair
that satisfy all the equations in the system.
A system of linear equations can have:
•1. Exactly one solution
•2. No solutions
•3. Infinitely many solutions
13
Inconsistent
Dependent
One solution
No solution
Lines intersect
Lines are parallel
Infinite number of
solutions
Consistent
Coincide-Same line
14
Types of Solutions
ONE Solution
There is only one answer that makes
the equation true
(EX: x=3 This is the answer)
Types of Solutions
NO Solution
There is no number that will satisfies
both sides of the equation. Variables will
cancel and a false statement is left.
(EX: 5=3 Answer: No solution)
Types of Solutions
Both lines are on
top of each other
MANY Solutions (Infinite)
Any number substituted for the
variable will make the equation true.
All variables will cancel and a true
statement will be left.
(EX: -2=-2 Answer: All real numbers)
Special Case #1
6) 2x + 5 = 2x - 3
-2x
-2x
1. Draw “the river”
5 = -3
2. Subtract 2x from
This is never true!
both sides
3. Simplify
No solutions
1.
2.
3.
4.
5.
Special Case #2
7) 3(x + 1) - 5 = 3x - 2
Draw “the river” 3x + 3 – 5 = 3x - 2
Distribute
3x - 2 = 3x – 2
Combine like
-3x
-3x
terms
Subtract 3x from
-2 = -2
both sides
This
is
always
true!
Simplify
Infinite solutions
or identity
Special Case #3
8) 3x + 4x – 4 = 10
7x – 4 = 10
7x = 10 + 4
7x = 14
x=2
ONE SOLUTION
What is the value of x if
-3 + 12x = 12x - 3?
1.
2.
3.
4.
0
4
No solutions
Infinite solutions
Answer Now
Challenge! What is the value of x if
-8(x + 1) + 3(x - 2) = -3x + 2?
1.
2.
3.
4.
-8
-2
2
8
Answer Now