Solving Linear Equations

To Solve an Equation means...

• To isolate the variable having a coefficient of 1 on one side of the equation.

Examples• x = 5 is solved for x.• y = 2x - 1 is solved for y.

Solving Equations Using Addition and Subtraction

Addition Property of Equality

For any numbers a, b, and c, if a = b, then a + c = b + c.

What it means:

You can add any number to BOTH sides of an equation and the equation will still hold true.

An easy example:

We all know that 7 =7.

Does 7 + 4 = 7? NO!

But 7 + 4 = 7 + 4.

The equation is still

true if we add 4

to both sides.

Let’s try another example!

x - 6 = 10

Add 6 to each side.

x - 6 = 10 +6 +6 x = 16

• Always check your solution!!

• The original problem is x - 6 = 10.

• Using the solution x=16,Does 16 - 6 = 10?

• YES! 10 = 10 and our solution is correct.

What if we see y + (-4) = 9?

Recall that y + (-4) = 9

is the same as y - 4 = 9.

Now we can use the addition property.

y - 4 = 9

+4 +4

y = 13

• Check your solution!

• Does 13 - 4 = 9?• YES! 9=9 and

our solution is correct.

How about -16 + z = 7?• Remember to always

use the sign in front of the number.

• Because 16 is negative, we need to add 16 to both sides.

• -16 + z = 7

+16 +16

z = 23

• Check you solution!

• Does -16 + 23 = 7?

• YES! 7 = 7 and our solution is correct.

A trick question...-n - 10 = 5

+10 +10

-n = 15• Do we want -n? NO,

we want positive n.• If the opposite of n

is positive 15, then n must be negative 15.

• Solution: n = -15

• Check your solution!• Does -(-15)-10=5?• Remember, two

negatives = a positive• 15 - 10 = 5 so our

solution is correct.

Subtraction Property of Equality

• For any numbers a, b, and c, if a = b, then a - c = b - c.

What it means:• You can subtract any number from

BOTH sides of an equation and the equation will still hold true.

3 Examples:1) x + 3 = 17

-3 -3

x = 14• Does 14 + 3 = 17?

2) 13 + y = 20

-13 -13

y = 7• Does 13 + 7 = 20?

3) z - (-5) = -13• Change this

equation. z + 5 = -13

-5 -5

z = -18• Does -18 -(-5) = -13?• -18 + 5 = -13• -13 = -13 YES!

Try these on your own...

x + 4 = -10 x – 14 = -5

y – (-9) = 4 3 – y = 7

12 + z = 15 -5 + z = -7

The answers...

x = -14 x = 9

y = -5 y = -4

z = 3 z = -2

Solving Equations Using Multiplication and Division

An easy example:

We all know that 3 = 3.

Does 3 4 = 3? NO!

But 3 4 = 3 4.

The equation is still true if we multiply both sides by 4.

Let’s try another example!

x = 4 2

Multiply each side by 2.

2 x = 4 2 2x = 8

• Always check your solution!!

• The original problem is

x = 4 2• Using the solution x = 8,

Is x/2 = 4?• YES! 4 = 4 and our

solution is correct.

A fraction times a variable:

The two step method:Ex: 2x = 4 31. Multiply by 3.(3)2x = 4(3) 32x = 12

2. Divide by 2.2x = 12 2 2x = 6

The one step method:

Ex: 2x = 4 31. Multiply by the

RECIPROCAL.

(3)2x = 4(3)(2) 3 (2)

x = 6

x

5

x5

x

5

• The two negatives will cancel each other out.

• The two fives will cancel each other out.

(-5) (-5)

• x = -15• Does -(-15)/5 = 3?

What do we do with negative fractions?

Recall that

Solve .

Multiply both sides by -5.

x5

3

x5

3

Try these on your own...

x = 3 7

4w = 16

y = 8 -2

2x = 12 3

-2z = -12 3x = 9 -4

Division Property of Equality

For any numbers a, b, and c (c ≠ 0), if a = b, then a/c = b/c

What it means: You can divide BOTH sides of an

equation by any number - except zero- and the equation will still hold true.

2 Examples:

1) 4x = 24

Divide both sides by 4.

4x = 24

4 4

x = 6

• Does 4(6) = 24?

YES!

2) -6x = 18

Divide both sides by -6.

-6y = 18

-6 -6

y = -3

• Does -6(-3) = 18? YES!

The answers...

x = 21 w = 4

y = -16 x = 18

z = 6 x = -12

Solving Equations with the Variable on Both Sides

To solve these equations,

•Use the addition or subtraction property to move all variables to one side of the equal sign.

•Solve the equation using the methods we mentioned.

Let’s see a few examples:

1) 6x - 3 = 2x + 13

-2x -2x

4x - 3 = 13

+3 +3

4x = 16

4 4

x = 4

Be sure to check your answer!

6(4) - 3 =? 2(4) + 13

24 - 3 =? 8 + 13

21 = 21

Let’s try another!

2) 3n + 1 = 7n - 5

-3n -3n

1 = 4n - 5

+5 +5

6 = 4n

4 4

Reduce! 3 = n

2

Check:

3(1.5) + 1 =? 7(1.5) - 5

4.5 + 1 =? 10.5 - 5

5.5 = 5.5

Here’s a tricky one!

3) 5 + 2(y + 4) = 5(y - 3) + 10

• Distribute first.

5 + 2y + 8 = 5y - 15 + 10• Next, combine like

terms.

2y + 13 = 5y - 5• Now solve. (Subtract

2y.)

13 = 3y - 5 (Add 5.)

18 = 3y (Divide by 3.)

6 = y

Check:

5 + 2(6 + 4) =? 5(6 - 3) + 10

5 + 2(10) =? 5(3) + 10

5 + 20 =? 15 + 10

25 = 25

Let’s try one with fractions!

4)

3 - 2x = 4x - 6

3 = 6x - 6

9 = 6x so x = 3/2

3

8

1

4x

1

2x

3

4Steps:• Multiply each termby the least common denominator (8) to eliminate fractions.

• Solve for x.• Add 2x.• Add 6.• Divide by 6.

(8)3

8 (8)

1

4x(8)

1

2x (8)

3

4

Two special cases:

6(4 + y) - 3 = 4(y - 3) + 2y

24 + 6y - 3 = 4y - 12 + 2y

21 + 6y = 6y - 12

- 6y - 6y

21 = -12 Never true!

21 ≠ -12 NO SOLUTION!

3(a + 1) - 5 = 3a - 2

3a + 3 - 5 = 3a - 2

3a - 2 = 3a - 2

-3a -3a

-2 = -2 Always true!

We write IDENTITY.

Try a few on your own:

• 9x + 7 = 3x - 5

• 8 - 2(y + 1) = -3y + 1

• 8 - 1 z = 1 z - 7

2 4

The answers:

• x = -2

• y = -5

• z = 20