Equations with Unknowns on Both Sides Slideshow 20, Mathematics Mr Richard Sasaki

Objectives Recall some algebraic vocabulary Solve equations where we must add and subtract unknowns within an equation Understand the number of solutions an equation has

3 𝑥 + 4 Vocabulary We have heard some of the following before… Constant (𝑥) Co-efficient Operator Terms A constant is a term where its value never changes (usually a single number).

Unknowns on Both Sides We know how to solve equations with constants on both sides of an equation. Example Solve 3𝑥+2=11. −2 −2 3𝑥=9 ÷3 ÷3 𝑥=3 How do we calculate with unknowns on both sides? Why did we subtract by 3𝑥, not 5𝑥? Example Solve 3𝑥+2=5𝑥. −3𝑥 −3𝑥 2=2𝑥 We should separate constants from unknowns. ÷2 ÷2 𝑥=1

Unknowns on Both Sides There are sometimes a few ways to calculate, it can take more steps if we avoid negative numbers. Example 6𝑥−16=2𝑥 Solve 6𝑥−16=2𝑥. +16 +16 −6𝑥 −6𝑥 −16=−4𝑥 6𝑥=2𝑥+16 −2𝑥 ÷(−4) ÷(−4) −2𝑥 4𝑥=16 𝑥=4 ÷4 ÷4 𝑥=4 Both layouts are fine. You should continue to lay things out like this! Don’t worry about writing the parts shown in blue!

Answers - Easy 𝑥=5 𝑥=4 𝑥=−2 2𝑥=6 4𝑥=16 3𝑥=18 𝑥=3 𝑥=4 𝑥=6 2𝑥=−4 −2𝑥=4 −𝑥=2 𝑥=−2 𝑥=−2 𝑥=−2 5𝑥=5 2𝑦=6 −𝑎=0 𝑥=1 𝑦=3 𝑎=0

Answers - Hard 9𝑦=72 4𝑎=96 −2𝑏=−52 𝑦=8 𝑎=24 𝑏=26 −2𝑐=32 −5𝑥=45 −𝑥=−1 𝑐=−16 𝑥=−9 𝑥=1 𝑥 2 =9 3𝑥 2 =75 7𝑥 2 =63 𝑥=±3 𝑥 2 =25 𝑥 2 =9 𝑥=±5 𝑥=±3 4𝑎 2 =9 𝑎 2 −405 −2𝑦 2 =−72 −4𝑎 2 =−196 −5 𝑎 2 =−405 𝑦 2 =36 𝑎 2 =49 𝑎 2 =81 𝑦=±6 𝑎=±7 𝑎=±9

How Many Solutions? So far, all equations we have solved have had one or two solutions. There are also equations with zero or infinite solutions. Note: There are also equations with three, four or another finite number of solutions. These are equations with unknowns that have greater powers. Usually equations with infinite solutions are called identities. We use the symbol ‘≡’ to show that two things are identical. For example… 𝑥 −1 ≡ 1 𝑥 .

Infinite Solutions Equations have ‘=’ symbols as we need to find out whether both sides are identical or not. Example Solve 𝑥+3−3=𝑥. Simplifying the left-hand side we get 𝑥=𝑥. This is true no matter the value of 𝑥 so there are infinite solutions. Example Solve 2 𝑥+1 =2𝑥+2. This is true always as 2(𝑥+1) is always equal to 2𝑥+2 so there are infinite solutions.

No Solutions Some equations have no solution. Still however, they will contain ‘=’ symbols! Example Solve 2𝑥+3=2𝑥. If we subtract 2𝑥 from each side we get . 3=0 Of course 3 and 0 are not equal so there are zero solutions. Example Solve 2 𝑥 2 =−2. If we divide by 2, we get 𝑥 2 =−1. No real number squared makes a negative so there are zero solutions.

Answers 1 solution 1 solution 0 solutions 𝑥=−1 𝑥=−1 Infinite solutions 𝑥=±2 1 solution 2 solutions 0 solutions 𝑥=−3 𝑥=±2 𝑥=0 As 𝑥=0, we cannot divide by 𝑥.