## Core concept 2.2:

## Solving linear equations

2.2.1.1 Recognise that there are many different types of equations of which linear is one type

2.2.1.2 Understand that in an equation the two sides of the 'equals' sign balance

2.2.1.3* Understand that a solution is a value that makes the two sides of an equation balance

2.2.1.4 Understand that a family of linear equations can all have the same solution

2.2.1.3 Understand that a solution is a value that makes the two sides of an equation balance

●Understand that the solution to an equation is a particular snapshot of a relationship between a variable and an expression.

●Understand that the solution to an equation is the value of the variable at which two expressions are balanced.

## Solving equations using dynamic bar models with variable x

The examples in this section allow you to find the value of x that makes the two sides of the equation equal in value.

Move the sliders to find the values of x that make the values of two expressions equal.

3x+1 = 13

3x - 7 = 8

-7 is represented as adding -7 on the bar model.

15 - x = 12

15-x is represented as -x+15

3x + 1 = 7x - 17

6x + 2 = -4x + 10

Click here to save your own copy of the equation variable slider tool for equations of the form px + ab = qx + d.

# The balance method

### Checkpoints activities

Checkpoints are diagnostic activities that will help teachers assess the understanding students have brought with them from primary school, and suggest ways to address any gaps that become evident.

https://www.ncetm.org.uk/classroom-resources/checkpoints/

2.2.3.1 Understand that an equation needs to be in a format to be 'ready' to be solved, through collecting like terms on each side of the equation

2.2.3.2 Know that when an additive step and a multiplicative step are required, the order of operations will not affect the solution

2.2.3.3* Recognise that equations with unknowns on both sides of the equation can be manipulated so that the unknowns are on one side

2.2.3.4 Solve complex linear equations, including those involving reciprocals

2.2.3.3 Recognise that equations with unknowns on both sides of the equation can be manipulated so that the unknowns are on one side

●Understand that an equation can be considered as both a process and an object.

●Understand ways in which equations can be manipulated while maintaining equality.

●Understand the manipulations that will efficiently work towards solving an equation.

## Solving equations using the balance method

Add x's, -x's, 1's and -1's to both sides of the equation solver on MathsBot

This model uses zero pairs rather than subtraction.

3x + 1 = 13

Add -1 to both sides

3x + 1 -1 = 13 -1

Simplify the expressions on both sides

3x = 12

Divide both sides by 3

x = 4

3x - 7 = 8

Add 7 to both sides

3x - 7 + 7 = 8 + 7

Simplify the expressions on both sides

3x = 15

Divide both sides by 3

x = 5

15 - x = 12

Add x to both sides

15 - x + x = 12 + x

Simplify the expressions on both sides

15 = 12 + x

Add -12 to both sides

15 - 12 = 12 + x -12

3 = x

3x + 1 = 7x - 17

Add -3x to both sides

3x + 1 - 3x = 7x - 17 - 3x

Simplify the expressions on both sides

1 = 4x - 17

Add 17 to both sides

1 + 17 = 4x -17 + 17

18 = 4x

Divide both sides by 4

4.5 = x

6x + 2 = -4x + 10

Add 4x to both sides

6x + 2 + 4x = -4x + 10 + 4x

Simplify the expressions on both sides

10x + 2 = 10

Add -2 to both sides

Divide both sides by 10

x = 0.8

The balance method using zero pairs leads to the written balance method.

Click here to see more examples of solving equations

## Bar model to graph

This model combines the bar model, table and graph.

Scroll down the control panel on the left to change the equation and switch off bar models.

Click here to save your own copy of the bar model/graph slider tool for equations of the form px + ab = qx + d.

2.2.4.1 Appreciate the significance of the bracket in an equation

2.2.4.2 Recognise that there is more than one way to remove a bracket when solving an equation

2.2.4.3 Solve equations involving brackets where simplification is necessary first