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DJB Lesson(key):
Alg-1A_One-Step Eqns(2.1)_L1
Alg 1A
2.1 - Solving One-Step Equations
Key Concepts
An equation is a mathematical sentence with an equal sign
Examples:
x + 2 = 11
2x - 8 = -18
3(x-2) = 7
To solve an equation with variables, we find the value(s) of the variable
that make the equation true. The value of the variable that makes
the equation true is called the solution of the equation.
We can think of the 2 sides of the equation like a balance.
 since both sides are equal, it is in balance
 if we add (or subtract) the same amount on both sides, it stays in balance
 if we take the same fraction (or multiple) of both sides, it stays in balance
Example
Starting Equation:
3x + 7 = 34
x
+
x
+
x
+
7
=
34
x
+
x
+
x
+
0
=
27
=
9
Subtract 7 from both sides:
3x+7 -7 = 34 -7
3x = 27
Multiply both sides by (1/3) (same as dividing by 3)
(1/3) * 3x = (1/3) * 27
x = 9
x
We follow a similar step-by-step approach when we solve algebraic equations,
keeping the equation in balance at each step, until we finally isolate the variable.
Steps for Solving an Equation (i.e., isolating the variable)
If applic., combine Like Terms, eliminate double-negatives,
5-
First simplify each side
apply the Distributive Property, etc.
Determine the inverse operation to be used
f ocus on the side with the variable
Perform the same inverse operation on both sides of the equation
Simplify both sides
Repeat, if needed, until the variable is completely isolated.
6-
Check the solution (substitute it for the variable in the original equation to verify
1234-
5-
Simplify both sides
Repeat, if needed, until the variable is completely isolated.
6-
Check the solution (substitute it for the variable in the original equation to verify
4-
that it makes the equation true).
Note: Each time we perform the same operation on both sides
of an equation, this results in a new but Equivalent Equation.
An Equivalent Equation has the same solution as the original equation.
e.g.,
3x + 7 = 34
3x = 27
x = 9
3(9) + 7 = 34
3(9) = 27
9 = 9
These are Equivalent equations
(all have the solution, x=9)
Inverse Operations
To solve an equation, we perform inverse (opposite) operations that
help to isolate the variable by undoing operations in the original equation.
Examples of Inverse Operations
Operation
+3
-7
*2
* (3/4)
/ 10
Inverse Operation
-3
same as + (-3)
+7
* (1/2) same as / 2
* (4/3) same as / (3/4)
* 10
Cancellation (Un-doing)
+3 -3 = 0
a dditive inverses
-7 +7 = 0
*2 *(1/2) = *1
multiplica tive
*(3/4)*(4/3)= *1
inverses
*(1/10) * 10 = *1
Identifying the inverse operation to be performed on both sides
Equivalent Equation when we
Given Equation
Operation
to be undone
Inverse
Operation
perform the Inverse Operation
on both sides of the equation
x + 5 = 22
+5
-5
x+5 - 5 = 22 - 5
x-7 =4
-7
+7
x - (3/4) = 4 1/2
- (3/4)
+ 3/4
x-(3/4) + 3/4 = 4 1/2 + 3/4
7x = -22
*7
* (1/7)
7x * (1/7) = -22 * (1/7)
(2/3)x = 16
* (2/3)
* (3/2)
(3/2) * (2/3)x = 16 * (3/2)
x / 5 = -25
/5
*5
5 * (x / 5) = -25 * 5
using
additive
inverses
x-7 +7 = 4 +7
Since we are trying to isolate the variable:
If we start with something added to x, we use the additive inverse
using
multiplicative
inverses
Since we are trying to isolate the variable:
If we start with something added to x, we use the additive inverse
If we start with something multiplied by x, we use the multiplicative inverse
Examples (via whiteboard):
After simplifying each side, determine the inverse operation to be used.
Then, solve the equation by performing the same operation on both sides
Cornell Notes format: For future practice, cover right side to hide answers; re-work problem & check .
(1)
x - 2.5 = 7
x - 2.5
=7
x - 2.5 + 2.5 = 7 + 2.5
x
<< add 2.5 to each side
= 9.5
Check:
(9.5) - 2.5
(2)
-x/2 = 7
= 7
-x/2 = 7
(-1/2)x = 7
(-2)* (-1/2)x = (-2) * 7
x
= -14
Check:
(-14) / (-2) = 7
(3)
n + 8 - 10 = -7
n + 8 - 10 = -7
n - 2
= -7
n - 2 + 2 = -7 + 2
n
<< first simplify left side
<< add 2 to both sides
= -5
Check:
(-5) + 8 - 10 = -7
(4)
(1/3)y = -27
(1/3)y = - 27
(3)* (1/3)y = (3) * (-27)
y
= -81
Check:
(1/3)(-81) = -27
(5)
(3/4)x = - 42
(3/4)x = - 42
(4/3)* (3/4)x = (4/3) * (-42)
x
= -56
Check:
(3/4)(-56) = -42
To Re-cap:
<< mult. by reciprocal of 3/4
(3/4)(-56) = -42
To Re-cap:
Steps
Helpful Tips for Solving Equations:
1. Write Eqn
•
Write each equation neatly & leave room
2. Simplify each side
•
Keep the equations neatly lined up on each side
3. Determine Inverse Op.
•
Re-write new simplified equations at each step
4. Perform on both sides
(6)
5/6x = -150
<< Hint: Focus on the side with the variable;
use the reciprocal to cancel out (undo) 5/6
(7)
8x + x + 9x = -36
<< Hint: first simplify left side by combining lik e terms
(8)
2 (32) * x = - 54
<< Hint: first simplify left side