Solving Linear Equations

4 + 1 = 13𝑥• What number would make this equation 𝑥

true? That is, what value of would make the 𝑥left side equal to the right side?

• In other cases, guessing the correct answer is not so easy. Consider the following equation:

3(4 − 9) + 10 = 15 + 2 + 7𝑥 𝑥 𝑥• Can you guess a number for that would 𝑥

make this equation true?

• Guessing is not always an efficient strategy for solving equations. In the last example, there are several terms in each of the linear expressions comprising the equation. This makes it more difficult to easily guess a solution.

• For this reason, we want to use what we know about the properties of equality to transform equations into equations with fewer terms

Solving Equations Using Algebra

• As you have seen from our visual models, we need to preserve balance on both sides of the equal sign.

• This is called the property of equality– When manipulating equations you must do the

same operation to both sides of the equal sign to preserve the balance of the equation.

Solve the linear equation 2 − 6 = 4x𝑥• Examine the properties of equality. Choose

“something” to add, subtract, multiply, or divide on both sides of the equation.

Using Algebra

x + 8 = 30

x + 7 = -3

3x = -18

𝑥3=−4

8n + 7 = 31

3p − 2 = −29

+ 5 = 3

−9 + = −7

Kind of Creepy

Combining Like Terms

𝒙 + + + + + + = −28𝒙 𝟐 𝒙 𝟒 𝒙 𝟔

r + 11 + 8r = 29

6r − 1 + 6r = 11

-6n − 2n = 16

The Distributive Property

3(x + 4) = (x + 4) + (x + 4) + (x + 4)

5(x + 3)

3(x – 5)

-4(x – 6)

37 = −3 + 5(x + 6)

10(1 + 3b) = −20

30 = −5(6n + 6)

−6(1 − 5v) = 54

8 = 8v − 4(v + 8)

−5n − 8(1 + 7n) = −8

−(1 + 7x) − 6(−7 − x) = 36

= 5

= 7

= 7

Variables on Both Sides

• Use the same process for solving equations, but now you must move variable terms all to one side of the equation using addition or subtraction

4x + 8 = 2x + 16

3x – 6 = 8x + 19

− 𝟐𝟗 𝟑𝒙 = + 5𝟓𝒙

5p − 14 = 8p + 4

−18 − 6k = 6(1 + 3k)

2(4x − 3) − 8 = 4 + 2x

− 𝑥 21 = 15

𝑥 + 13 + = 1 − 9 + 22𝑥 𝑥

𝟐( + ) = − + x𝟑𝒙 𝟐 𝟐𝒙 𝟏

− 𝒙 𝟗 = x

− 𝒙 𝟓 + = 𝟏𝟕𝟏 x