Equations with Numbers and

Unknowns on Both Sides

Slideshow 21, Mathematics

Mr Richard SasakiRoom 307

Objectives• Review algebraic vocabulary from

last lesson• Solve by forming the unknown on one

side of the equation and a value on the other side

• Recall how to expand brackets and use this to solve such equations

• Consider numbers of solutions

Vocabulary ReviewDo you remember this from last time?

Terms Co-efficient Operator

Constant

3𝑥+¿4Please try to get familiar with this!The expression above is linear.This means all terms have unknowns to the power 0 or 1.

Unknowns on Both SidesLet’s review solving for unknowns on both sides from last lesson.ExampleSolve −5 𝑥 −5 𝑥−2 𝑥=−6÷(−2) ÷(−2)𝑥=3We try to get the unknown amount on one side and the constant number on the other side, even if we end up with negative numbers.

Numbers and Unknowns on Both SidesThe principle for equations with numbers and unknowns on both sides is the same. We separate the equation so numbers are on one side and algebraic terms are on the other side.ExampleSolve .We have 4 options of subtraction, 2 of which are more sensible. What are the two best things we could choose to do?It is best to either or . Why?

−2 −3 𝑥

Numbers and Unknowns on Both SidesExampleSolve .If we choose to or , we will avoid negative numbers. Of course other options are okay too.5 𝑥+2=3 𝑥+8−2 −25 𝑥=3 𝑥+6−3 𝑥 −3 𝑥2 𝑥=6÷2 ÷2𝑥=3

5 𝑥+2=3 𝑥+8−3 𝑥 −3 𝑥2 𝑥+2=8−2 −22 𝑥=6÷2 ÷2𝑥=3

Any order is okay!

Answers - Easy𝑥=2 𝑥=5 𝑥=3

𝑥=4 𝑥=−4 𝑥=16

𝑦=1 𝑥=−1 𝑥=−5𝑥=3 𝑥=1 𝑥=6

𝑥=9 𝑦=− 12 𝑥=0

Answers - Hard𝑥=±2𝑥=±1 𝑥=± 12

𝑥=±5𝑥=±2 𝑥=± 4𝑥=±3𝑥=0 𝑥=± 4𝑥=−2𝑥=2𝑥=− 75 𝑜𝑟 𝑥=5

Expanding BracketsLet’s review how to expand brackets. For linear style brackets, we multiply the term on the outside with the terms on the inside.

7 (3 𝑥+4)¿21 𝑥+28We will use this concept to solve equations.ExampleSolve .6 𝑦−8=20−8 𝑦+8 𝑦 +8 𝑦14 𝑦−8=20+8 +814 𝑦=28÷14 ÷14𝑦=2

Expanding BracketsLet’s try a harder question.Solve .12𝑥+21=(36 𝑥+54)−2112𝑥   +  21  =   36 𝑥  +   54  −  2112𝑥   +  21  =  36 𝑥  +  33−12 𝑥 −12 𝑥21=24 𝑥+33−33 −33−12=24 𝑥÷(−24) ÷(−24)𝑥=− 12

Answers𝑥=4 𝑥=5

𝑥=2 𝑥=19

𝑥=6 𝑥=4

𝑥=−2 𝑥=4

Number of SolutionsWhen we try to solve an equation, we may find that it has solutions or infinite solutions.𝑥+3=𝑥 solutions 𝑥+2𝑥=3 𝑥Infinite solutions

ExampleState the number of solutions has.

3 (𝑥+2 )=3𝑥−53 𝑥+6=3 𝑥−5

6=−5As both sides do not equate, there are solutions.

Answers

Infinite solutions solutions

Infinite solutions solutions

solution, solutions, or