SOLVING EQUATIONS

• The single most important skill in algebra

• Goal: Find the one value of the variable that makes the sentence true.

For instance, if 2x + 3 = 17,

then 7 is the only value of x that will make this a true sentence.

So x = 7.

• We can solve equations by doing the OPPOSITE of what

has been done to the variable in the problem.

• If a problem says +, you subtract.

• If a problem has multiplication, you divide.

By doing the opposite, we keep the sides of the equation balanced.

As long as you do the SAME thing to both sides of an equation, it will remain balanced.

7x – 13 = 50

7x – 13 = 50+13 +13

7x = 63

7x – 13 = 50+13 +13

7x = 63 7 7 x = 9

-5x + 7 = 82

-5x + 7 = 82- 7 - 7

-5x = 75

-5x + 7 = 82- 7 - 7

-5x = 75-5 -5 x = -15

4x + 25 = 13

4x + 25 = 13- 25 -25

4x = -12

4x + 25 = 13- 25 -25

4x = -12 4 4 x = -3

When you solve equations, you also do the opposite of the order of operations.

Add/subtract first

Then divide/multiply

Solve these equations:

4a + 11 = 59

-2b + 13 = 5

5c – 72 = 98

-4d + 11 = 47

Solve these equations:

4a + 11 = 59 a = 12

-2b + 13 = 5 b = 4

5c – 72 = 98 c = 34

-4d + 11 = 47 d = -9

Solve these equations:

5x – 18 = 40

2y + 73 = 54

-3z + 5 = 1

Solve these equations:

5x – 18 = 40 x = 58/5 or 11.6

2y + 73 = 54 y = -19/2 or -9.5 _

-3z + 5 = 1 z = 4/3 or 1.3

Some equations are even easier.

n + 4 = 13

5x = 35

Some equations are even easier.

n + 4 = 13Just subtract 4 … x = 9

5x = 35Just divide by 5 … x = 7

Solve

17 = 3x – 7

Solve

17 = 3x – 7+7 +724 = 3x

Solve

17 = 3x – 7+7 +724 = 3x 3 3 8 = x

Solve

19 – 2x = 104

Solve

19 – 2x = 104-19 -19 -2x = 85

Solve

19 – 2x = 104-19 -19 -2x = 85 -2 -2 x = -85/2 or -42.5

If you know the basic steps, you can quickly do equations with more difficult numbers using a calculator.

12x + 1794 = 2127

12x + 1794 = 2127

963 – 25x = 704

963 – 25x = 704

What about this?

22137

n

Fractions mean division, so to cancel, we’ll subtract 13 and then multiply by 7.

n = 63

22137

n

Things that can complicate solving equations …

Parentheses• Use distributive property

first.Like terms• Combine them first.

3(2x – 5) = 27

3(2x – 5) = 27

3(2x – 5) = 27 6x – 15 = 27 6x = 42 x = 7

-7(2x – 11) = 98

-7(2x – 11) = 98 -14x + 77 = 98 -14x = 21 x = -3/2 or -1.5

4p + 3 – 2p + 7 + 5p + 2 = 17

4p + 3 – 2p + 7 + 5p + 2 = 17 7p + 12 = 17 7p = 5 p = 5/

7

5(3x + 5) – 3(2x – 1) = 145

5(3x + 5) – 3(2x – 1) = 145 15x + 25 – 6x + 3 = 145

5(3x + 5) – 3(2x – 1) = 145 15x + 25 – 6x + 3 = 145 9x + 28 = 145

5(3x + 5) – 3(2x – 1) = 145 15x + 25 – 6x + 3 = 145 9x + 28 = 145 9x = 117 x = 13

The goal is always to simplify.

Make the problem look like the easy ones we know how to solve.

Variable on Both Sides• Find the smaller number of

the variable, and subtract that on both sides.

• Solve the remaining problem.

7x – 15 = 2x + 70

7x – 15 = 2x + 70-2x -2x5x – 15 = 70

7x – 15 = 2x + 70-2x -2x5x – 15 = 705x = 85 x = 17

5x + 13 = 7x + 40

5x + 13 = 7x + 40-5x -5x 13 = 2x + 40

5x + 13 = 7x + 40-5x -5x 13 = 2x + 40 x = -27/2 or -13.5

3(2x + 7) = 3x + 4 + x + 9

3(2x + 7) = 3x + 4 + x + 9 6x + 21 = 4x + 13

3(2x + 7) = 3x + 4 + x + 9 6x + 21 = 4x + 13 2x + 21 = 13 2x = -8 x = -4

Yesterday you saw this equation:

3x + 5(4x – 2) – 2(x – 4) = 4(2x – 7 + 3x)

Solve it.

3x + 5(4x – 2) – 2(x – 4) = 4(2x – 7 + 3x)3x + 20x – 10 – 2x + 8 = 8x – 28 + 12x

3x + 5(4x – 2) – 2(x – 4) = 4(2x – 7 + 3x)3x + 20x – 10 – 2x + 8 = 8x – 28 + 12x 21x – 2 = 20x – 28

3x + 5(4x – 2) – 2(x – 4) = 4(2x – 7 + 3x)3x + 20x – 10 – 2x + 8 = 8x – 28 + 12x 21x – 2 = 20x – 28-20x -20x x – 2 = -28 +2 +2 x = -26

Special equations

2(3x – 7) = 6x + 11

10x – 15 = 5(2x – 3)

2(3x – 7) = 6x + 11

6x – 14 = 6x + 11

?????

10x – 15 = 5(2x – 3)

10x – 15 = 10x – 15

?????

When variables cancel out…• If you have the exact

same thing on both sides (like 8 = 8), the answer is ALL REAL NUMBERS or INFINITELY MANY SOLUTIONS.

10x – 15 = 5(2x – 3)10x – 15 = 10x – 15

-15 = -15

• An equation with infinitely many solutions can also be called an IDENTITY.

• If there is something different on the 2 sides

(like 5 = 7), there is NO SOLUTION.

2(3x – 7) = 6x + 116x – 14 = 6x + 11

-14 = 11

• If there is something different on the 2 sides

(like 5 = 7), there is NO SOLUTION.

2(3x – 7) = 6x + 116x – 14 = 6x + 11

-14 = 11