multi step equation PP

To solve
• multi-step equations,
• equations with no solutions,
• equations with an infinite

number of solutions

Course 3, Lesson 2-5

Expressions and Equations

Number of Solutions

Course 3, Lesson 2-5

Expressions and Equations

Null Set One Solution Identify

Words no solution one solution infinitely many solutions

Symbols a = b

x = a a = a

Example 3x + 4 = 3x 2x =20 4x + 2 = 4x + 2
4 = 0 x = 10 2 = 2
Since 4 ≠ 0, Since 2 = 2, the

there is no solution is all
solution. numbers.

Need Another Example?

Step-by-Step Example

1.

Solve 15(20 + d) = 420.

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15(20 + d) = 420

Write the equation.

Subtraction Property of Equality

Simplify.

300 + 15d = 420

Distributive Property

Simplify.

15d = 120

Division Property of Equality

d = 8

– 300 = – 300

Answer

Need Another Example?

Solve –3(4p – 6) = 54.

–3

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Need Another Example?

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Step-by-Step Example

2.

Solve 6(x – 3) + 10 = 2(3x – 4).

6(x – 3) + 10 = 2(3x – 4)

Write the equation.

Collect like terms.

Simplify.

6x – 18 + 10 = 6x – 8

Distributive Property

6x – 8 = 6x – 8

Addition Property of Equality

6x = 6x

Simplify.

Division Property of Equality

x = x

The statement x = x is always true. The equation is an identity
and the solution set is all numbers.

Write the original equation.

Check 6(x – 3) + 10 = 2(3x – 4)

Substitute any value for x.

6(5 – 3) + 10 = 2[3(5) – 4]

Simplify.

6(2) + 10 = 2(15 – 4)

22 = 22

?

?

+ 8 = + 8

Answer

Need Another Example?

Solve 3(4x + 8) = 2(6x + 12).

identity; all numbers

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Need Another Example?

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Step-by-Step Example

3.

Solve 8(4 – 2x) = 4(3 – 5x) + 4x.

8(4 – 2x) = 4(3 – 5x) + 4x

Write the equation.

Collect like terms.

Simplify.

32 – 16x = 12 – 20x + 4x

Distributive Property

32 – 16x = 12 – 16x

Addition Property of Equality

32 = 12

The statement 32 = 12 is never true. The equation has no
solution and the solution set is .

Write the equation.

Check8(4 – 2x) = 4(3 – 5x) + 4x

Substitute any value for x.

8[4 – 2(2)] = 4[3 – 5(2)] + 4(2)

Simplify.

0 ≠ –20

?

8(0) = 4(–7) + 8

?

Since 0 ≠ –20, the equation has no solution.

+ 16x = + 16x

Answer

Need Another Example?

Solve 4(5x + 3) – 6x = 7(2x + 3).

null set; no solution

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Need Another Example?

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Step-by-Step Example

4.

At the fair, Hunter bought 3 snacks and 10 ride tickets.
Each ride ticket costs $1.50 less than a snack. If he spent
a total of $24.00, what was the cost of each snack?

Write an equation to represent the problem.

Write the equation.

Simplify.

3s + 10(s – 1.5) = 24

Addition Property of Equality

13s = 39

3s + 10s – 15 = 24

Distributive Property

13s – 15 = 24

Collect like terms.

Division Property of Equality

Simplify.

s = 3

So, the cost of each snack was $3.

+ 15 = + 15

Answer

Need Another Example?

The length of Philip’s stride when walking is 4 inches
greater than the length of Anne’s stride. If it takes Philip
5 steps and Anne 6 steps to walk the same distance,
what is the length of Anne’s stride?

20 in.

Course 3, Lesson 2-1

Describe how the previous
lesson on solving equations
with variables on each side

helped you with today’s

lesson on solving multi-step

equations.

Ratios and Proportional Relationships

Expressions and Equations

Course 3, Lesson 2-5