Algebraic Equations
Ambar Mitra
Equation
On the see-saw shown in the picture below, we have loaded a sack full of sugar on
one side, and a carton of milk on the other.
When the see-saw is level, we know
The weight of the sack of sugar is the same as the weight of the carton of milk.
In algebra, instead of the phrase “is the same as”, we use the word “equals.”
The weight of the sack of sugar equals the weight of the carton of milk.
In algebra, we shorten what we write by using some symbols.
Symbol for weight of sack of sugar is “S”
Symbol for weight of carton of milk is “M”
Symbol for “equals” is “=”, and this symbol is called the “equal” sign.
Therefore, for the level see-saw, we write
S=M
What you see above is called an “Equation”
The word “equation” is derived from the word “equal.”
For now, we will talk about level see-saws and equations only.
The most important rule for an equation is
Whatever you do to the left side of the equation,
You must do the same to the right side of the equation.
Only when you follow this rule, the see-saw remains level.
If you double the left side, you must double the right side.
The equation for this picture is
2S = 2M
If you halve the left side, you must halve the right side.
The equation for this picture is
½S=½M
If you add 4 ounces of sugar in a bowl to the left side, you must add 4 ounces of
milk in a bowl to the right side.
The equation for this picture is
S+4=M+4
Solving Equations
Example 1
The see-saw below has a sack of sugar and 4 ounces of sugar in a bowl on one side.
We made the see-saw level by putting 76 ounces of weight on the other side. What
is the weight of one sack of sugar?
In the style of algebra we write
S + 4 = 76
Remember, the most important rule of algebra
Whatever you do to the left side of the equation,
You must do the same to the right side of the equation.
By subtracting 4 ounces from both sides of the equation, we get
S = 72
The answer to our problem is – a sack of sugar weighs 72 ounces.
Example 2
The see-saw below has two sacks of sugar and 4 ounces of sugar in a bowl on one
side. We made the see-saw level by putting 160 ounces of weight on the other
side. What is the weight of one sack of sugar?
In the style of algebra we write
2S + 4 = 160
Remember, the most important rule of algebra
Whatever you do to the left side of the equation,
You must do the same to the right side of the equation.
By subtracting 4 ounces from both sides of the equation, we get
2S = 156
By dividing both sides of the equation by 2, we get
S = 78
The answer to our problem is – a sack of sugar weighs 78 ounces.
Example 3
The see-saw below has a two sacks of sugar and 4 ounces of sugar in a bowl on one
side. We made the see-saw level by putting one sack of sugar and 60 ounces of
weight on the other side. What is the weight of one sack of sugar?
In the style of algebra we write
2S + 4 = S + 60
Remember, the most important rule of algebra
Whatever you do to the left side of the equation,
You must do the same to the right side of the equation.
By subtracting 4 ounces from both sides of the equation, we get
2S = S + 56
By subtracting S from both sides of the equation, we get
S = 56
The answer to our problem is – a sack of sugar weighs 56 ounces.
Example 4
The see-saw below has one sack of sugar, one-half sack of sugar, and 4 ounces of
sugar in a bowl on one side. We made the see-saw level by putting one sack of
sugar and 40 ounces of weight on the other side. What is the weight of one sack of
sugar?
In the style of algebra we write
½ S + S + 4 = S + 40
Remember, the most important rule of algebra
Whatever you do to the left side of the equation,
You must do the same to the right side of the equation.
By subtracting 4 ounces from both sides of the equation, we get
½ S + S = S + 36
By subtracting S from both sides of the equation, we get
½ S = 36
By multiplying both sides of the equation by 2, we get
S = 72
The answer to our problem is – a sack of sugar weighs 72 ounces.
Example 5 (with substitution when two pieces of information are given)
Three bags of sugar and one carton of milk weigh 280 ounces.
3S + M = 280
From this first piece of information, we find
M = 280 – 3S first information
Four bags of sugar and two cartons of milk weigh 400 ounces.
4S + 2M = 400 second information
Substituting first information in second information we find
4S + 2(280 – 3S) = 400
Or
4S + 560 – 6S = 400
Or
-2S + 560 = 400
Or
-2S = 400 – 560 = -160
Or
S = 80
Weight of one sack of sugar is 80 ounces.