At the end of the lesson, the students should be able to solve linear equations involving one unknown.
A linear equation is any equation that can be written in the form ax+b=0.
Where a and b are real numbers and is a variable. This form is sometimes called the standard form of a linear equation. Note that most linear equations will not start off in this form. Also, the variable may or may not be an x so don’t get too locked into always seeing an x there. Variable(s) in linear expressions:
Cannot have exponents (or powers).For example, x squared or x2.
Cannot be found under a root sign or square root sign. For example: √x or the “square root x”.
Cannot multiply or divide each other. For example: “x” times “y” or xy; “x” divided by “y” or x/y.
Here is a simple example: x + 2 = 10.
“A certain number, plus 2, equals 10.”
An equation has two sides: the left side, x + 2, and the right side, 10.
An equation is a statement that two quantities are equivalent.
The basic approach to finding the solution to equations is to change the equation into simpler equations, but in such a way that the solution set of the new equation is the same as the solution set of the original equation. When two equations have the same solution set, we say that they are equivalent.
What we want to do when we solve an equation is to produce an equivalent equation that tells us the solution directly. Going back to our previous example,
“x + 2 = 10“, we can say that x = 8
We solve equations by using methods that rearrange the equation in a manner that does not change the solution set, with a goal of getting the variable by itself on one side of the equal sign. The solution is just the number that appears on the other side of the equal sign. The important thing to remember about any equation is that the equals sign represents a balance.
What an equals sign says is that what’s on the left-hand side is exactly the same as what’s on the right-hand side. So, if we do anything to one side of the equation we have to do it to the other side. If we don’t, the balance is disturbed. Therefore, whatever operation we perform on either side of the equation, so long as it’s done in exactly the same way on each side, the balance will be preserved.
Process for Solving Linear Equations
The process of finding out the value of the variable that makes the equation true is called ‘solving’ the equation.
The equation will become true only when the unknown has a certain value, which we call the solution to the equation.
Now, algebra depends on how things look. As far as how things look, then, we will know that we have solved an equation when we have isolated x on the left.
Why the left? Because it is easier to read, from left to right. “x equals . . .”
To solve a linear equation we can use:
Inverse law method (METHOD 1 ).
the principles of addition/subtraction, multiplication or division (METHOD 2).
The law of inverses
There are two pairs of inverse operations. Addition/subtraction and multiplication /division.
We shift a number from one side of an equation to the other by writing it on the other side with the inverse operation.
1)If x + a = b, then x = b − a.
“If a number is added on one side of an equation, we subtract it from the other side.”
For example x +6 = 8
Implies x= 8 – 6
2) If x − a = b, then x = b + a.
“If a number is subtracted on one side of an equation, we add it to the other side.“
For example x -6 = 8
Implies x= 8 +6
3) If ax = b, then x = b/a
“If a number multiplies one side of an equation, we divide it by the other side.”
2x = 6
2x /2= 6/2
4) If x/a = b, then x =ab
“If a number divides one side of an equation,we may multiply it by the other side.”
For example x/2= 6
x = 6×2
(METHOD 2 )
The Addition and subtraction Principle
1) If a=b then a +c=b+c for any c. All this is saying is that we can add a number, c, to both sides of the equation and it will not change the equation.
2) If a=b then a – c=b -c for any c. We can subtract a number, c, from both sides of an equation.
Adding (or subtracting) the same number to both sides of an equation does not change its solution set.
Think of the balance analogy—if both sides of the equation are equal, then increasing both sides by the same amount will change the value of each side, but they will still be equal. For example, if
3 = 3,
Then 3 + 2 = 3 + 2.
Consequently, if 6 + x = 8
For some value of x (which in this case is x = 2), then we can add any number to both sides of the equation and x = 2 will still be the solution. If we wanted to, we could add a 3 to both sides of the equation, producing the equation
6 + x+ 3 = 8+ 3
9 + x = 11.
As you can see, x = 2 is still the solution.
The Multiplication and division principle
3) If a = b then ac=bc for any c. Like addition and subtraction we can multiply both sides of an equation by a number, c, without changing the equation.
4) If a =b then a/c=b /c for any non-zero c. We can divide both sides of an equation by a non-zero number, c, without changing the equation.
Multiplying (or dividing) the same non-zero number to both sides of an equation does not change its solution set.
3×6×2=12 × 3
So if 6x = 12,
Then 18x = 36
This gives the same value of x (which in this case is x = 2).
2x = 6
These facts form the basis of almost all the solving techniques that we’ll be looking at in this unit so it’s very important that you know them and don’t forget about them. As long as you always do the same thing to BOTH sides of the equation, and do the operations in the correct order, you will get to the solution.
Solve for ‘x’:
1)a) x+ 1 = 3 b) 2x = 12
1)a) x+ 1 = 3
x + 1 = 3 Subtract 1 from the right side.
x=3 – 1 Simplify right sides.
x = 2
x + 1 = 3 Subtract 1 from both sides.
x + 1 – 1 = 3 -1 Simplify both sides.
x = 2
1)b) –2x= 12
–2x = 12 Divide right sides by -2.
x = 12/-2 Simplify.
x = – 6
–2x = 12 Divide both sides by -2
–2x /-2 = 12 /-2 Simplify both sides:
x = –6
Using the operation together
Suppose you were given an equation like
2x – 3 = 5.
You will need to use the addition principle to move the –3, and the multiplication principle to remove the coefficient 2. Which one should you use first? Strictly speaking, it does not matter—you will eventually get the right answer. In practice, however, it is usually simpler to use the addition principle first, and then the multiplication principle. The reason for this is that if we divide by 2 first we will turn everything into fractions:
Given: 2x – 3 = 5
Suppose we first divide both sides by 2:
(2x – 3)/2 = 5/2
x – 3/2 = 5/2
Now there is nothing wrong with doing arithmetic with fractions, but it is not as simple as working with whole numbers. In this example we would have to add 3/2 to both sides of the equation to isolate the x.
x = 5/2 + 3/2
x = 8/2
x = 4
It is usually more convenient, though, to use the addition/subtraction operation first.
Given: 2x – 3 = 5 Add 3 to the right side
2 x=5+3 Simplify
2x=8 Divide both side by 2.
Solve for ‘x’: 2x + 1 = 17
2x + 1 = 17 Subtract 1 from right side.
2x=17 –1 Simplify.
2x=16 Divide the sides by 2.