Solving Linear Equations
Study Cards
A linear equation is a pattern
of numbers that have
proportional increase and
decrease which is used to
represent a line graph
for example y = 2x + 1
What is a linear equation?
1.Expand all brackets
2. Move all terms with variables on one
side of the equation & all constant terms
on the other side
3. Group like terms together & simplify
them
4. Factorize if needed
General Steps
(equation) true.
Perform
operations on
both sides of the
equation to
isolate the
variable.
Goal
Method
The goal of solving a
linear equation is to
find the value of the
variable that will
make the statement
6. Check the answer by substituting the
solution back into the original equation
5. Find the solution & write down the
steps
NOTE: What we do to one side of an
equation we must also do to the other side
of the equation.
General steps continued
Let a, b, and c represent
algebraic expressions.
1.Addition property of equality:
If a = b,
then a + c = b + c
2. Subtraction property of
equality:
If a = b
then a - c =b - c
Example 1: Addition
1.Add to both sides
x-3 =7
10 - 3 = 7
7= 7
+3 + 3
x = 10
Example 2: Subtraction
2.Subtract 5 from both sides
x + 5 =-2
- 5 -5
x= -7
Addition and Subtraction
Properties of Equality
Solution check
Solution check
(-7) + 5 =-2
-2 = -2
Addition and Subtraction
Properties of Equality continued
Let a, b, and c represent algebraic
expressions.
1.Multiplication property of equality:
if a = b
then ac = bc
2. Division property of equality:
if a = b
then a/c = b/c (provided c = 0)
Example 1: Multiplication
a/-7 = -42
Multiply both sides by -7.
-7 (a/-7) = -7 (-42)
Simplify
a = 294
Check your answer
a/-7 = -42
Multiplication and Division of
Properties of equality
Let a = 294
294/-7 = -42
-42 = -42
Example 2: Division
4x = -28
Divide both sides by 4 to undo the
multiplication.
4x/4 = -28/4
Simplify
x = -7
Check your answer
4x = -28
Let x = -7. Substitute -7 for x.
4 (-7) = -28
-28 = -28
Multiplication and Division of
Properties of equality continued
When solving equations with fractions or
decimals, clear the fractions or decimals in
order to make the equation simpler
1.To clear fractions, multiply both sides of
the equation to distribute all terms by the
LCD of all the fractions
2.To clear decimals multiply both sides of
the equation to distribute all terms by the
lowest power of 10 that will make ALL
decimals whole decimals whole numbers
Example 1: Fractions
1/8x + 1/2 = 1/4
LCD = 8
Multiply both sies of the equation by
the LCD, 8 to clear the fractions.
8( 1 x + 1 ) = 8( 1 )
8
2
4
Clearing Fractions and Decimals
in an Equation
Use the distributive property
8 1x + 8 1 =8 1
8
2
4
Simplify
x +4 = 2
Solve
x+4-4=2-4
Simplify
x = -2
Check
1x+1=1
8
2 4
1 (-2) + 1 = 1
8
2 4
-2 + 1 = 1
8
2 4
Clearing Fractions and Decimals
in an Equation continued
2=1
8 4
1=1
4 4
Example 2: Decimals
0.8x - 5 = 7
Multiply both sides by the LCD
10 (0.8x - 5 = 10(7)
Distribute
10 (0.8x) - 10 (5) = 10 (7)
Multiply
8x - 50 = 70
Add 50 to get all constants to the
right
8x - 50 + 50 = 70 + 50
Clearing Fractions and Decimals
in an Equation continued
Simplify
8x = 120
Divide both sides by 8
8x = 120
8
8
Simplify
x = 15
Check
0.8 (15) - 5 = 7
12 - 5 = 7
7=7
Clearing Fractions and Decimals
in an Equation continued
When solving an equation, if the
variables are eliminated to reveal a
true statement such as, -13 = -13, then
the solution is all real numbers. This
type of equation is called an identity.
On the other hand, if the variables are
eliminated to reveal a false statement
such as -7 = 3, then there is no
solution. This type of equation is
called a contradiction. All other linear
equations which have only one
solution are called conditional.
Example 1: Identity
2x + 6 = 3 (x +2) -x
Expand brackets
2x + 6 = 3x + 6 - x
Subtract from both sides
2x + 6 = 2x + 6
True statement
-2
-2
Solution: all real numbers
6=6
Example 2: Contradiction
5x - 3 = 4 (x+ 2) + x
Expand brackets
5x -3 = 4x + 8 + x
Subtract from both sides
5x - 3 = 5x + 8
-5
-5
False statement
-3=8
No solution
Note: Identity Vs Contradiction
Note: Identity Vs Contradiction
continued