# Building Expressions Using Backtracking

Lesson

To construct equations, we need to think about how our unknown value has been changed.

1. The unknown may be multiplied or divided by a number, or both.
2. There may be a value added or subtracted
3. Brackets may be placed around an expression

So, to get back to our original unknown value (and solve the equation), we need to be like detectives and backtrack through these steps by using inverse (or opposite) operations. To do this, we need to consider the order of operations.

Consider the equation $2x-1=3$2x1=3

Starting with $x$x, we get to a value of $3$3 by:

1. Multiplying by $2$2

2. Then subtracting $1$1

To solve for $x$x, we need to reverse each operation in reverse order!

Remember:

• The inverse or opposite of addition is subtraction.
• The inverse or opposite of multiplication is division.

#### Examples

##### Question 1

Solve the following equation using backtracking: $x-16=14$x16=14

Think: Let's build up an expression starting from $x$x and construct a flowchart using inverse operations to solve the equation.

Do:

 $x-16$x−16 $=$= $14$14 (add $16$16 to both sides) $x$x $=$= $30$30

##### Question 2

Solve the following equation using backtracking: $5x+4=34$5x+4=34

Think: Let's build up an expression starting from $x$x and construct a flowchart using inverse operations to solve the equation.

Do:

 $5x+4$5x+4 $=$= $34$34 (subtract $4$4 from both sides) $5x$5x $=$= $30$30 (divide both sides by $5$5) $x$x $=$= $6$6

##### Question 3

By working backwards in single steps, complete the entries.

### Outcomes

#### NA4-7

Form and solve simple linear equations