Solving Multi-Step Equations

A multi-step equation requires more than one operation to isolate the variable.

For example:

3x + 5 = 20

The variable x is first multiplied by 3 and then 5 is added.

To solve the equation, we undo these operations in the reverse order:

3x + 5 = 20

Subtract 5:

3x = 15

Divide by 3:

x = 5

Multi-step equations build directly on the same balance and inverse-operation ideas used for one-step equations.

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The Goal: Isolate the Variable

To isolate the variable means to get the variable by itself on one side of the equation.

For:

4x − 7 = 21

we want to reach:

x = ?

There are two operations affecting x:

×4

and:

−7

We undo them systematically.

Add 7:

4x = 28

Divide by 4:

x = 7

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Keep the Equation Balanced

An equation represents two equal quantities.

Whatever valid operation we perform on one side must also be performed on the other side.

For example:

2x + 6 = 18

Subtract 6 from both sides:

2x + 6 − 6 = 18 − 6

2x = 12

Then divide both sides by 2:

2x/2 = 12/2

x = 6

The equation remains balanced throughout the process.

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Inverse Operations

The main inverse operations are:

Multi-step equations usually require us to use more than one of these.

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Undo Operations in Reverse Order

Consider:

5x + 8 = 33

To create the left side:

1. x was multiplied by 5.
2. Then 8 was added.

To solve, reverse that order:

1. subtract 8
2. divide by 5

5x + 8 = 33

5x = 25

x = 5

This is sometimes described as undoing the operations from the outside inward.

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Why Reverse Order?

Suppose:

3x + 4 = 19

The expression 3x + 4 means:

Start with x.

↓

Multiply by 3.

↓

Add 4.

To get back to x:

Subtract 4.

↓

Divide by 3.

This reverses the original sequence.

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A Reliable Solving Strategy

For many multi-step equations, use this sequence:

Simplify → Move → Isolate → Check

Simplify

• distribute if necessary
• combine like terms

Move

• move variable terms to one side if needed
• move constants to the other side

Isolate

• use inverse operations until the variable is alone

Check

• substitute the answer into the original equation

This method works for a wide range of equations.

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Two-Step Addition Equation

Solve:

4x + 7 = 31

Subtract 7:

4x = 24

Divide by 4:

x = 6

Check:

4(6) + 7 = 31

24 + 7 = 31

31 = 31

Therefore:

x = 6

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Two-Step Subtraction Equation

Solve:

5x − 9 = 26

Add 9:

5x = 35

Divide by 5:

x = 7

Check:

5(7) − 9 = 26

35 − 9 = 26

26 = 26

Correct.

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Division and Addition

Solve:

x/4 + 3 = 8

Subtract 3:

x/4 = 5

Multiply by 4:

x = 20

Check:

20/4 + 3 = 8

5 + 3 = 8

Correct.

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Division and Subtraction

Solve:

x/5 − 4 = 3

Add 4:

x/5 = 7

Multiply by 5:

x = 35

Check:

35/5 − 4 = 3

7 − 4 = 3

Correct.

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Negative Coefficients

Solve:

−3x + 5 = 20

Subtract 5:

−3x = 15

Divide by −3:

x = −5

Check:

−3(−5) + 5 = 20

15 + 5 = 20

Correct.

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Negative Constants

Solve:

4x − 6 = −18

Add 6:

4x = −12

Divide by 4:

x = −3

Check:

4(−3) − 6 = −18

−12 − 6 = −18

Correct.

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Equations with Several Terms

Sometimes we must simplify before isolating the variable.

Consider:

3x + 5 + 2x = 25

Combine like terms:

5x + 5 = 25

Subtract 5:

5x = 20

Divide by 5:

x = 4

Check:

3(4) + 5 + 2(4) = 25

12 + 5 + 8 = 25

25 = 25

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Combining Like Terms

Like terms have the same variable raised to the same power.

For example:

3x + 5x = 8x

and:

7 − 2 = 5

But:

3x + 5

cannot be combined because one term contains x and the other does not.

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Worked Example: Combine First

Solve:

7x − 3x + 6 = 30

Combine like terms:

4x + 6 = 30

Subtract 6:

4x = 24

Divide by 4:

x = 6

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The Distributive Property

Some equations contain parentheses.

For example:

3(x + 4) = 24

We can use the distributive property:

3x + 12 = 24

Subtract 12:

3x = 12

Divide by 3:

x = 4

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Distributive Property

The distributive property states:

a(b + c) = ab + ac

and:

a(b − c) = ab − ac

For example:

4(x + 3)

becomes:

4x + 12

Similarly:

5(x − 2)

becomes:

5x − 10

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Worked Example: Distribution

Solve:

2(x + 5) = 18

Distribute:

2x + 10 = 18

Subtract 10:

2x = 8

Divide by 2:

x = 4

Check using the original equation:

2(4 + 5) = 18

2(9) = 18

18 = 18

Correct.

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Negative Distribution

Be especially careful when distributing a negative number.

Solve:

−2(x − 3) = 10

Distribute −2:

−2x + 6 = 10

Subtract 6:

−2x = 4

Divide by −2:

x = −2

Check:

−2(−2 − 3) = 10

−2(−5) = 10

10 = 10

Correct.

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Variables on Both Sides

Some equations contain variables on both sides.

For example:

5x + 2 = 3x + 14

We first collect variable terms on one side.

Subtract 3x from both sides:

2x + 2 = 14

Subtract 2:

2x = 12

Divide by 2:

x = 6

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Which Variable Term Should You Move?

Consider:

2x + 15 = 7x

We could subtract either 2x or 7x from both sides.

It is often convenient to move the smaller variable term:

Subtract 2x:

15 = 5x

Divide by 5:

3 = x

Usually we write:

x = 3

Choosing a convenient side can reduce negative numbers, but either correct method works.

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Worked Example: Variables on Both Sides

Solve:

8x − 5 = 3x + 20

Subtract 3x:

5x − 5 = 20

Add 5:

5x = 25

Divide by 5:

x = 5

Check:

Left side:

8(5) − 5 = 35

Right side:

3(5) + 20 = 35

Therefore:

35 = 35

Correct.

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Variables and Negative Numbers

Solve:

3x + 10 = 7x − 6

Subtract 3x:

10 = 4x − 6

Add 6:

16 = 4x

Divide by 4:

x = 4

Check:

LHS:

3(4) + 10 = 22

RHS:

7(4) − 6 = 22

Correct.

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Parentheses on Both Sides

Solve:

2(x + 3) = 4(x − 2)

Distribute:

2x + 6 = 4x − 8

Subtract 2x:

6 = 2x − 8

Add 8:

14 = 2x

Divide by 2:

x = 7

Check:

LHS:

2(7 + 3) = 20

RHS:

4(7 − 2) = 20

Correct.

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Simplify Both Sides First

Consider:

3(x + 2) + x = 2x + 14

Distribute:

3x + 6 + x = 2x + 14

Combine like terms:

4x + 6 = 2x + 14

Subtract 2x:

2x + 6 = 14

Subtract 6:

2x = 8

Divide by 2:

x = 4

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A More Complex Example

Solve:

4(2x − 3) + 5 = 3x + 18

Distribute:

8x − 12 + 5 = 3x + 18

Combine constants:

8x − 7 = 3x + 18

Subtract 3x:

5x − 7 = 18

Add 7:

5x = 25

Divide by 5:

x = 5

Check:

LHS:

4(2(5) − 3) + 5

= 4(10 − 3) + 5

= 4(7) + 5

= 33

RHS:

3(5) + 18

= 15 + 18

= 33

Correct.

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Equations Containing Fractions

Fractions can make an equation look more difficult, but the same principles apply.

Solve:

x/3 + 4 = 10

Subtract 4:

x/3 = 6

Multiply by 3:

x = 18

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Clearing Fractions

Sometimes several fractions appear in an equation.

For example:

x/2 + x/3 = 10

The denominators are 2 and 3.

Their lowest common multiple is:

6

Multiply every term by 6:

6(x/2) + 6(x/3) = 6(10)

Simplify:

3x + 2x = 60

5x = 60

x = 12

Check:

12/2 + 12/3 = 10

6 + 4 = 10

Correct.

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Equations with Decimals

Solve:

0.5x + 3 = 9

Subtract 3:

0.5x = 6

Divide by 0.5:

x = 12

Check:

0.5(12) + 3 = 9

6 + 3 = 9

Correct.

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Clearing Decimals

Sometimes it is convenient to multiply the entire equation by a power of 10.

Consider:

0.2x + 1.5 = 3.1

Multiply every term by 10:

2x + 15 = 31

Subtract 15:

2x = 16

Divide by 2:

x = 8

Check:

0.2(8) + 1.5 = 3.1

1.6 + 1.5 = 3.1

Correct.

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Equations with a Fractional Coefficient

Solve:

(3/4)x + 2 = 11

Subtract 2:

(3/4)x = 9

Multiply by the reciprocal 4/3:

x = 9 × 4/3

x = 12

Check:

(3/4)(12) + 2 = 11

9 + 2 = 11

Correct.

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One Solution

Most equations in this topic have exactly one solution.

Example:

3x + 4 = 19

Solving gives:

x = 5

There is exactly one value that makes the equation true.

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No Solution

Some equations have:

no solution

Consider:

3(x + 2) = 3x + 10

Distribute:

3x + 6 = 3x + 10

Subtract 3x:

6 = 10

This is false.

Therefore:

no value of x can satisfy the equation

So the equation has:

no solution

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Infinitely Many Solutions

Consider:

4(x + 2) = 4x + 8

Distribute:

4x + 8 = 4x + 8

Subtract 4x:

8 = 8

This statement is always true.

Therefore every permissible value of x satisfies the equation.

The equation has:

infinitely many solutions

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Recognizing the Three Outcomes

After simplifying, you may obtain:

Variable remains

2x = 10

This gives:

one solution

False statement

4 = 9

This means:

no solution

True statement

7 = 7

This means:

infinitely many solutions

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Verifying Solutions

Checking becomes especially important with multi-step equations because there are more opportunities for mistakes.

Suppose:

2(3x − 1) = 22

Solve:

6x − 2 = 22

6x = 24

x = 4

Now check using the original equation:

2(3(4) − 1) = 22

2(12 − 1) = 22

2(11) = 22

22 = 22

The solution is verified.

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Why Check the Original Equation?

Suppose you make a mistake while simplifying.

If you check only your simplified equation, you might not detect that mistake.

Substituting into the:

original equation

tests whether the proposed solution truly satisfies the equation you were asked to solve.

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Explaining Your Reasoning

A strong solution does more than give the answer.

Consider:

6x − 8 = 28

A clear explanation would be:

"First, I add 8 to both sides because addition is the inverse of subtracting 8. This gives 6x = 36. Then I divide both sides by 6 because x is multiplied by 6. This gives x = 6. Finally, I substitute 6 into the original equation and obtain 28 = 28, so the solution is correct."

This explains:

• the operation
• the inverse operation
• why both sides are changed
• the solution
• the verification

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Writing Clear Algebraic Work

Instead of jumping directly from:

5x + 7 = 32

to:

x = 5

show the important steps:

5x + 7 = 32

5x = 25

x = 5

Clear working makes it easier to:

• explain reasoning
• find mistakes
• check calculations
• communicate mathematics

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Multi-Step Equations from Word Problems

Equations can model real situations.

Example

A taxi charges a fixed fee of $4 plus $3 per kilometer.

A journey costs $25.

Let x be the number of kilometers.

Equation:

3x + 4 = 25

Subtract 4:

3x = 21

Divide by 3:

x = 7

The journey was:

7 km

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Worked Word Problem: Shopping

A person buys 4 identical notebooks and pays a $6 delivery charge.

The total cost is $38.

Let x represent the cost of one notebook.

4x + 6 = 38

Subtract 6:

4x = 32

Divide by 4:

x = 8

Each notebook costs:

$8

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Worked Word Problem: Temperature

A temperature in Fahrenheit can be related to Celsius using:

F = (9/5)C + 32

Suppose:

F = 68

Then:

68 = (9/5)C + 32

Subtract 32:

36 = (9/5)C

Multiply by 5/9:

C = 20

Therefore:

68°F = 20°C

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Multi-Step Equations in Science

Many scientific calculations involve multi-step equations.

For example:

v = u + at

Suppose:

v = 25 m/s

u = 5 m/s

t = 4 s

Then:

25 = 5 + 4a

Subtract 5:

20 = 4a

Divide by 4:

a = 5 m/s²

The same algebraic methods apply.

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Another Science Example

Suppose:

E = mcΔT

and:

E = 8400 J

m = 2 kg

c = 420 J/(kg·°C)

Then:

8400 = 2(420)ΔT

8400 = 840ΔT

Divide by 840:

ΔT = 10°C

Solving scientific equations relies heavily on algebraic isolation.

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Reverse Operations and Order of Operations

When evaluating an expression, we use the order of operations.

For example:

3x + 5

means:

1. multiply x by 3
2. add 5

When solving:

3x + 5 = 20

we undo those operations in reverse:

1. subtract 5
2. divide by 3

So solving can be thought of as:

reversing the construction of the expression

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A Useful Flow for More Complex Equations

For equations such as:

3(2x + 5) − 4 = 2x + 27

use:

Distribute

↓

Combine like terms

↓

Collect variable terms

↓

Collect constants

↓

Isolate the variable

↓

Check

Let's apply it.

Distribute:

6x + 15 − 4 = 2x + 27

Combine:

6x + 11 = 2x + 27

Subtract 2x:

4x + 11 = 27

Subtract 11:

4x = 16

Divide by 4:

x = 4

Check:

LHS:

3(2(4) + 5) − 4

= 3(13) − 4

= 39 − 4

= 35

RHS:

2(4) + 27

= 35

Correct.

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Common Mistakes

Mistake: Undoing operations in the wrong order.

For:

3x + 6 = 18

remove the +6 before dividing by 3.

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Mistake: Performing an operation on only one side.

Equality must be maintained.

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Mistake: Forgetting to distribute to every term.

Incorrect:

3(x + 4) = 3x + 4

Correct:

3(x + 4) = 3x + 12

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Mistake: Incorrectly distributing a negative.

Correct:

−2(x − 5) = −2x + 10

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Mistake: Combining unlike terms.

3x + 5

cannot become:

8x

because 3x and 5 are unlike terms.

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Mistake: Moving terms without understanding the operation.

Terms do not magically "cross the equals sign and change signs."

For example:

3x + 7 = 19

becomes:

3x + 7 − 7 = 19 − 7

The −7 comes from subtracting 7 from both sides.

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Mistake: Losing negative signs.

Always carry signs carefully through each step.

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Mistake: Stopping too early.

If you reach:

4x = 20

the variable is not yet isolated.

Continue:

x = 5

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Mistake: Forgetting to check.

Substitute the final value into the original equation.

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Key Terms

Multi-step equation — equation requiring more than one operation to solve.

Variable — letter or symbol representing an unknown or changing value.

Solution — value that makes an equation true.

Inverse operations — operations that undo one another.

Isolate — get the variable alone on one side of an equation.

Like terms — terms containing the same variables raised to the same powers.

Combine like terms — simplify an expression by adding or subtracting coefficients of like terms.

Distributive property — property represented by a(b + c) = ab + ac.

Coefficient — numerical factor multiplying a variable.

Constant — number without a variable.

Substitution — replacing a variable with a particular value.

Verification — checking that a solution makes the original equation true.

Equivalent equations — equations having the same solution set.

Identity — equation true for every permissible value of the variable.

No solution — situation in which no value makes an equation true.

Reciprocal — number obtained by reversing the numerator and denominator of a nonzero fraction.

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Key Takeaways

• Multi-step equations require several operations to isolate a variable.
• The fundamental goal is to get the variable alone.
• Equality must be maintained throughout the solution.
• Inverse operations undo operations affecting the variable.
• Operations are often undone in reverse order.
• Simplify expressions before trying to isolate the variable.
• Use the distributive property to remove parentheses when needed.
• Combine like terms whenever possible.
• Variables appearing on both sides can be collected onto one side.
• Constants can then be collected onto the opposite side.
• Fractions can sometimes be removed by multiplying by a common denominator.
• Decimals can sometimes be removed by multiplying by a power of 10.
• Negative signs require careful attention.
• A multi-step equation may have one solution, no solution, or infinitely many solutions.
• A false statement such as 4 = 9 indicates no solution.
• A true statement such as 7 = 7 after the variable disappears indicates infinitely many solutions.
• Solutions should be checked by substitution into the original equation.
• Clear algebraic steps make mathematical reasoning easier to communicate and verify.
• Multi-step equations are important for solving mathematical models and rearranging equations in science.
• Systematic methods reduce mistakes and make even complicated equations manageable.

Check Your Understanding

1. What is a multi-step equation?
2. What does it mean to isolate a variable?
3. Why are operations often undone in reverse order when solving equations?
4. Solve: 3x + 5 = 20
5. Solve: 4x − 7 = 29
6. Solve: x/5 + 3 = 9
7. Solve: −3x + 8 = 23
8. Solve: 5x + 7 + 2x = 42
9. Solve: 3(x + 4) = 27
10. Solve: −2(x − 5) = 18
11. Solve: 5x + 3 = 2x + 21
12. Solve: 7x − 4 = 3x + 20
13. Solve: 2(x + 5) = 4(x − 1)
14. Solve: x/2 + x/3 = 15
15. Solve: 0.4x + 2.6 = 8.2
16. Determine whether 4(x + 2) = 4x + 8 has one solution, no solution, or infinitely many solutions.
17. Determine whether 3(x + 1) = 3x + 8 has one solution, no solution, or infinitely many solutions.
18. A taxi charges $5 plus $4 per kilometer. The total fare is $37. Write and solve an equation to determine the distance traveled.
19. A student solves 4x + 8 = 28 and obtains x = 9. Use substitution to determine whether the answer is correct and explain the error.
20. Explain a systematic method you could use to solve an equation containing parentheses, like terms, and variables on both sides.