Solving Linear Equations
To grasp a good understanding of linear equations let’s explore the following example;
I am thinking of a number. I multiply it by 3 and then subtract 1 from it. The final answer that I would get is a 14. What number am I thinking of?
The problem is asking to find a number which might seem impossible to find because we know the answer and not how it was formed. The number was multiplied by 3 then 1 was taken away to get 14 as shown below.
With this problem we can work backwards/reverse and undo the entire process as shown below;
Notice above that the signs have been changed to their opposite. You must know that the opposite of subtract is addition and the opposite of multiplication is division. I this problem when we add 1 to 14 and then divide by 3 we get;
In mathematics we use letters to represent unknown numbers. Instead of writing;
“I am thinking of a number and multiply it by 3 and subtracting 1 from it the answer is 14”
We could write;
We can also get rid of the multiplication sign.
So instead of writing 3 x x we write 3x so we write the problem as;
Letters to represent unknown numbers
To solve the problem we undo the problem as described earlier. Let the number be x. We know that the number x is multiplied by 3 and then 1 is subtracted from it. The answer found is 14. This is very basic to convert into algebra.
We carry through as we did in the previous example. We have to find the value of x. This process is very much similar to rearranging equations by making sure that the value of x is by itself on one side. There 2 basic steps when solving this kind of equation.
When you add 1 to both sides you get rid of the 1 from the left hand side of the equation.
The multiplication by 3 has to be undone by dividing both sides by 3.
…and…
Therefore dividing both sides by 3 should result;
Adding 3 to both sides of the equation will make sure that both sides are equal to each other while getting rid of 3 on the left hand side of the equation.
Now divide both sides by 6 to get rid of 6 on the left hand side so that x is left by itself.
To get rid of 2 from the left hand side of the equation subtract 2 from both sides.
To make sure that x is left alone on one side we divide both sides by 5.
X on both sides
Often you may get problems where x or unknown value appears on both sides, for example;
We can put the problem into algebra. Let x be the number that we’re trying to find.
The letter x has been used in the equation to represent unknown values. Again we follow a few basic steps.
By taking 4x from both sides of the equation we make sure that the term with the letter x appears on one side of the equation.
We must make sure that only x is left on the left hand side of the equation. Therefore we take 7 from both sides;
The term with x has a coefficient of 2. We must get rid of this. To do this we must divide both sides by 2;
Poof
We can test each side of the equation in turn to see whether we get the same answer for the x value that we have found. The left hand side of the equation is;.
…and the right hand side of the equation is;.
We can start with the left and see what answer we get when we insert the x values.
The answer we found for the left hand side of the equation was 43. Now we can try the right hand side as well.
The answer we get for the right hand side of the equation is also 43 when the found x value is used. Since the answer is the same in both sides that is clear proof that the value of x that we have found is correct.
In this example x appears on both sides of the equation.
Again we follow a few basic steps as well divide before.
We need to make sure that the term with x appears alone on one side of the equation. We do this by adding 4x to both sides of the equation.
The last step made sure that x appeared on the left hand side of the equation.
There is still a term (6) of the left and side which does not contain x, we have to get rid of this. To get rid of 6 from the left hand side we must subtract 6 from both sides of the equation
After subtracting 6 from both sides you will get the following expression.
To make sure that the x is left on the left hand side by itself we must divide both sides by 12. This should provide the value of x.
