Download Solving Equations with Variables on Both Sides

Goal:
I will solve linear equations in one variable.
❖
Linear equations in one variable with one solution,
infinitely many solutions, or no solutions.
❖
Linear equations with rational number coefficients,
including equations whose solutions require
expanding expressions using the distributive property
and combining like terms.
Solving 2 Step Equations - Let’s Review
1) GOAL:Get the variable on one side of the
equation.
2) You always perform the same operation to
both sides of an equation.
3) To undo an operation, you perform its
opposite operation to both sides of the
equation.
4) You should undo addition or subtraction
before you undo multiplication or division.
Solving 2 Step Equations
2x + 4 = 10
-4 -4
2x = __
__
6
2
2
x
=3
2(3) +4 = 10
6 + 4 = 10
10 = 10
Get the variable alone.
Subtract 4 from both sides.
+4 and -4 cancel out
Divide both sides by 2
2 divided by 2 cancel to 1
Show your check.
Finish your check.
Solving 2 Step Equations
x
3
3
x
3
x
(3
x
36
3
–4=8
+4 +4
Get the variable alone.
= 12
+ 4 and -4 cancel out.
Add 4 to both sides.
) = 3 12 Multiply both sides by 3.
= 36
–4=8
12 – 4 = 8
8=8
3 divided by 3 cancel to 1
Show your check.
Finish your check.
Try These Examples
1. 2x – 5 = 17
2. 3x + 7 = 25
3. x - 3 = 8
5
4. x + 4 = 9
9
Don’t Call Me Stupid…
D – Distribute
C – Combine like terms
M – Move the variable to one side
S – Solve
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.
•
•
•
•
•
•
1) Solve 8x - 9 = -3x + 2
+ 3x
+ 3x
Draw “the river”
11x – 9 =
2
Add 3x to both sides
Simplify
+9
+9
Add 9 to both sides
Simplify
11x = 11
Divide both sides by
11
11
11
Simplify
Check your answer
x=1
8(1) - 9 = -3(1) + 2
1.
2.
3.
4.
5.
6.
2) Solve 3x + 2 = 4x - 1
- 3x
- 3x
Draw “the river”
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
Multi-Step Equations: Get Variables on One Side
Try these on your own…
2x – 14 = 3x – 13
Multi-Step Equations: Get Variables on One Side
Try these on your own…
5y + 7 = 13 + 3y
Multi-Step Equations: Get Variables on One Side
Try these on your own…
11x + 30 = 5x – 42
Multi-Step Equations: Get Variables on One Side
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)
Solve Equations by Combining Like Terms
18 = 8x + x
2x – 4x + 2 = 22
Solve Equations by Combining Like Terms
-x + 7x + 1 = 37
10x – 6x = 2x + 14
Solve Equations by Combining Like Terms
1.
2.
3.
4.
5.
6.
7.
3) 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
Solve Equations by Using the
-7(1) = -7
Distributive Property
What is the value of x if
3(x + 4) = 2(x - 1)?
1.
2.
3.
4.
-14
-13
13
14
Answer Now
Solve Equations by Using the Distributive Property
Give it a shot…
3(x + 3) = 18
-4(y – 2) = 24
Solve Equations by Using the Distributive Property
4(2x – 5) + 4 = x + 5
Solve Equations by Using the Distributive Property
Putting it all together
3(x – 4) + 4x = 12 + x
Remember: D C M S !!!
Practice… Don’t Call Me Stupid
• Write the letters you use to solve the
problem. Which letter will always be used?
1) 3x – 14 = 46
• 1/3x – 4.1 = 6
• 3(3x – 3) = 27
Remember: D C M S !!!
4) 12.2x – 16.7 = 1.4x + 37.3
5) 9(x – 3) + 4 = 49 + 3x
Remember: D C M S !!!
Challenge! What is the value of x if
-8(x + 1) + 3(x - 2) = -3x + 2?
Remember: D C M S !!!
Balancing Scales
1. These scales are all currently balanced. You must choose a number to fill into
the boxes in each problem that will keep them balanced.
2. Whatever number you choose for a problem, you may only use that number.
Special Case #1
6) 2x + 5 = 2x - 3
1. Draw “the river”
2. Subtract 2x from both
sides
3. Simplify
-2x
-2x
5 = -3
This is never true!
No solutions
Remember: D C M S !!!
SPECIAL CASES
2x + 3 = 2x + 7
SPECIAL CASES
3(x + 4) = 3x + 11
Special Case #2
7) 3(x + 1) - 5 = 3x - 2
3x + 3 – 5 = 3x - 2
3x - 2 = 3x – 2
-3x
-3x
-2 = -2
This is always true!
Infinite solutions
Remember: D C Mor
S !!!identity
Draw “the river”
Distribute
Combine like terms
Subtract 3x from both
sides
5. Simplify
1.
2.
3.
4.
SPECIAL CASES
2x + 3 = 2x + 3
SPECIAL CASES
-2(x + 3) = -2x - 6
What is the value of x if
-3 + 12x = 12x - 3?
1.
2.
3.
4.
0
4
No solutions
Infinite solutions
Answer Now
Remember: D C M S !!!