Equations » Linear equations (Balance method)

Contents

1. What is a linear equation?
2. What is the balance method?
3. Examples

1. What is a linear equation?

If you only have linear formulas in your equation, you have a linear equation. When you have the variable on only one side of the =-sign, you can use the reversed arrow chain, card method or the balance method. When you have the variable on both sides of the =-sign, then you HAVE TO use the balance method.

Examples
3a + 5 = 26
5w + 7 = 4 – 8w
3x = 5x + 2x + 5


2. What is the balance method?

The balance method consist of only one rule:

Do the same operation on both sides of =-sign in such a way you are left with the unknown variable.

You remember it? Go directly to the examples.
Need more explanation? Read on below.

Extended explanation

With the balance method you can solve equations by doing the same operation on both sides of the =-sign. The name comes from the thought behind it:
A balance is an (old) weighing instrument with a scale/dish/weighing pan on both sides. On both sides of the balance you put a part of the equation. What is to the left of the =-sign in the equation goes on the left-hand scale and what is to the right of the =-sign in the equation goes on the right-hand scale. Now you start removing (or adding) things from (or to) the scale until you are left with the answer.

Example

Given is the equation 4a + 11 = 6a + 3.
You can make the following balance.
Take bags of marbles for a and separate marbles for the numbers.
Balance with on the left 4 bags and 11 marbles and on the right 6 bags and 3 marbles
On both sides you have separate marbles.
This means that on both sides you can remove three marbles without losing the equilibrium of the balance (the balance remains balanced).

You will get: 4a + 8 = 6a
Balance with on the left 4 bags and 8 marbles and on the right 6 bags and 0 marbles
You can also remove 4 bags of marbles without losing the equilibrium of the balance.

You will get: 8 = 2a
Balance with on the left 0 bags and 8 marbles and on the right 2 bags and 0 marbles
When two bags are equal to 8 marbles, there must be 8 : 2 = 4 marbles in each bag.
Solution: a = 4

The example and explanation above you have to keep in mind when solving any equation.
Not only linear equations. You have to use the balance method with other equations too.

3. Examples


8a + 4 = 5a + 16
–4 –4
8a = 5a + 12
–5a –5a
3a = 12
:3 :3
a = 4
3w + 7 = 9w – 23
–7 –7
3w = 9w – 30
–9w –9w
–6w = –30
: –6 : –6
w = 5


To do the first step in the equations below, you have to be able to simplify formulas.

15x – 4x + 5 = 12x + 4 – 3x
11x + 5 = 9x + 4 
–5 –5 
11x = 9x – 1 
–9x –9x 
2x = –1 
:2 :2 
x = 12 
–4(x + 5) = 3x – 11x – 8
–4x – 20 = –8x – 8 
+20 +20  
–4x = –8x + 12 
+8x +8x 
4x = 12 
:4 :4 
x = 3 


2(x – 6) = 5 – 3x
2x – 12 = 5 – 3x
+12 +12
2x = 17 – 3x
+3x +3x
5x = 17
:5 :5
x = 325
12(x – 2) = 23x – 5
------------------------ × 6
3(x – 2) = 4x – 30 
3x – 6 = 4x – 30 
+6 +6 
3x = 4x – 24 
–4x –4x 
x = –24 
: –1 : –1 
x = 24 


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