## How to Solve Multi Step Equations?

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Multi-step equations are equations that have to be solved by two consequent steps. Here we have to apply at least two mathematical operations and then only, we can find the value of the unknown variable.

### Properties of Multi-Step Equations

There are several properties of multi-step equations. They are:

• In a multi-step equation, we can add or subtract both sides of the equation by the same number. Also, this won’t affect the equation in any way and the equation would still hold true.
• Secondly, we can multiply or divide both sides of the equation by the same number. This too won’t affect the equation and the equation will still hold true.

### Solving Multi-Step Equations

For solving multi-step equations, follow these steps:

• For solving multi-step equations, you need to combine all like terms together.
• Then, you need to bring all variables to one side by either adding or subtracting.
• Next, simplify using the inverse of addition or subtraction.
• Next, you can further simplify by using the inverse of multiplication or division.

Example 1: $$7x \ + \ 15 \ = \ -3x \ + \ 9$$
Now, adding $$+3x$$ to both sides we get, $$7x \ + \ 3x \ + \ 15 \ = \ -3x \ + \ 3x \ + \ 9 \ ⇒ \ 10x \ = \ -6 \ ⇒ \ x \ = \ \frac{-6}{10}$$ .

Example 2: $$2x \ + \ 7 \ = \ -6x \ + \ 14$$
Now, adding $$+6x$$ to both sides we get, $$2x \ + \ 6x \ + \ 7 \ = \ -6x \ + \ 6x \ + \ 14 \ ⇒ \ 8x \ = \ 7 \ ⇒ \ x \ = \ \frac{7}{8}$$ .

Example 3: $$3x \ + \ 8 \ = \ -7x \ + \ 18$$
Now, adding $$+7x$$ to both sides we get, $$3x \ + \ 7x \ + \ 8 \ = \ -7x \ + \ 7x \ + \ 18 \ ⇒ \ 10x \ = \ 10 \ ⇒ \ x \ = \ \frac{10}{10} \ = \ 1$$.

Example 4: $$7x \ + \ 12x \ - \ 15 \ = \ 8x \ + \ 21$$
Now, subtracting $$8x$$ to both sides we get, $$7x \ + \ 12x \ – \ 8x \ - \ 15 \ = \ 8x \ - \ 8x \ + \ 21 \ ⇒ \ 11x \ = \ 36 \ ⇒ \ x \ = \ \frac{36}{11}$$ .

Example 5: $$-2x \ + \ 3 \ = \ 3x \ + \ 7$$
Now, adding $$2x$$ to both sides we get, $$2x \ - \ 2x \ + \ 3 \ = \ 3x \ + \ 2x \ + \ 7 \ ⇒ \ 3 \ = \ 5x \ + \ 7 \ ⇒ \ x \ = \ -\frac{4}{5}$$ .

### Exercises for Multi Step Equations

1) $$3x \ - \ 20 \ + \ 3x = 100 \ + \ x$$$$\ \Rightarrow \$$

2) $$6x \ - \ 16 \ + \ 3x = 40 \ + \ x$$$$\ \Rightarrow \$$

3) $$5x \ - \ 14 \ + \ 3x = 84 \ + \ x$$$$\ \Rightarrow \$$

4) $$\frac{ 5x \ - \ 24}{5} = 6 \ - \ \frac{1}{5}x$$$$\ \Rightarrow \$$

5) $$\frac{ 3x \ - \ 16}{7} = 32 \ - \ \frac{1}{7}x$$$$\ \Rightarrow \$$

6) $$\frac{ 5x \ - \ 24}{6} = 90 \ - \ \frac{1}{6}x$$$$\ \Rightarrow \$$

7) $$4x \ - \ 18 = 84 \ - \ 2x$$$$\ \Rightarrow \$$

8) $$5x \ - \ 21 = 49 \ - \ 2x$$$$\ \Rightarrow \$$

9) $$5x \ - \ 21 = 56 \ - \ 2x$$$$\ \Rightarrow \$$

10) $$5x \ - \ 14 = 105 \ - \ 2x$$$$\ \Rightarrow \$$

1) $$3x \ - \ 20 \ + \ 3x = 100 \ + \ x$$$$\ \Rightarrow \ \color{red}{3x \ + \ 3x \ - \ x = 100 \ + \ 20 }$$$$\ \Rightarrow \ \color{red}{5x = 120}$$$$\ \Rightarrow \ \color{red}{x = \frac{ 120}{5} = 24}$$
2) $$6x \ - \ 16 \ + \ 3x = 40 \ + \ x$$$$\ \Rightarrow \ \color{red}{6x \ + \ 3x \ - \ x = 40 \ + \ 16 }$$$$\ \Rightarrow \ \color{red}{8x = 56}$$$$\ \Rightarrow \ \color{red}{x = \frac{ 56}{8} = 7}$$
3) $$5x \ - \ 14 \ + \ 3x = 84 \ + \ x$$$$\ \Rightarrow \ \color{red}{5x \ + \ 3x \ - \ x = 84 \ + \ 14 }$$$$\ \Rightarrow \ \color{red}{7x = 98}$$$$\ \Rightarrow \ \color{red}{x = \frac{ 98}{7} = 14}$$
4) $$\frac{ 5x \ - \ 24}{5} = 6 \ - \ \frac{1}{5}x$$$$\ \Rightarrow \ \color{red}{5x \ - \ 24= 6 \times 5 \ - \ \frac{5}{5}x}$$$$\ \Rightarrow \ \color{red}{5x \ + \ x = 30 \ + \ 24}$$$$\ \Rightarrow \ \color{red}{x = \frac{ 30 \ + \ 24}{6} = 9}$$
5) $$\frac{ 3x \ - \ 16}{7} = 32 \ - \ \frac{1}{7}x$$$$\ \Rightarrow \ \color{red}{3x \ - \ 16= 32 \times 7 \ - \ \frac{7}{7}x}$$$$\ \Rightarrow \ \color{red}{3x \ + \ x = 224 \ + \ 16}$$$$\ \Rightarrow \ \color{red}{x = \frac{ 224 \ + \ 16}{4} = 60}$$
6) $$\frac{ 5x \ - \ 24}{6} = 90 \ - \ \frac{1}{6}x$$$$\ \Rightarrow \ \color{red}{5x \ - \ 24= 90 \times 6 \ - \ \frac{6}{6}x}$$$$\ \Rightarrow \ \color{red}{5x \ + \ x = 540 \ + \ 24}$$$$\ \Rightarrow \ \color{red}{x = \frac{ 540 \ + \ 24}{6} = 94}$$
7) $$4x \ - \ 18 = 84 \ - \ 2x$$$$\ \Rightarrow \ \color{red}{4x \ + \ 2x = 84 \ + \ 18 }$$$$\ \Rightarrow \ \color{red}{6x = 102}$$$$\ \Rightarrow \ \color{red}{x = \frac{ 102}{6} = 17}$$
8) $$5x \ - \ 21 = 49 \ - \ 2x$$$$\ \Rightarrow \ \color{red}{5x \ + \ 2x = 49 \ + \ 21 }$$$$\ \Rightarrow \ \color{red}{7x = 70}$$$$\ \Rightarrow \ \color{red}{x = \frac{ 70}{7} = 10}$$
9) $$5x \ - \ 21 = 56 \ - \ 2x$$$$\ \Rightarrow \ \color{red}{5x \ + \ 2x = 56 \ + \ 21 }$$$$\ \Rightarrow \ \color{red}{7x = 77}$$$$\ \Rightarrow \ \color{red}{x = \frac{ 77}{7} = 11}$$
10) $$5x \ - \ 14 = 105 \ - \ 2x$$$$\ \Rightarrow \ \color{red}{5x \ + \ 2x = 105 \ + \ 14 }$$$$\ \Rightarrow \ \color{red}{7x = 119}$$$$\ \Rightarrow \ \color{red}{x = \frac{ 119}{7} = 17}$$

## Multi Step Equations Practice Quiz

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