An equation is solved in two ways: numerically or symbolically. In numerical methods, only numbers are admitted as solutions. Whereas in the symbolical method expressions can be used for representing the solutions. There can be more than one solution for an equation. Solving a linear equation with one variable gives you a unique solution, solving a linear equation involving two variables gives you two results. An equation is a mathematical statement with two equal sides separated by an equal sign. Solving an equation means we are finding the values of the unknown variables that make the equation true. I gave 2 apples to Vedh and 2 to Vihan. I ask them to find the total number of apples. So they added the apples 2 + 2 = 4. Here 2 + 2 = 4 is an equation.
Definition of Linear Equation
Linear equations give a straight line when the two variables of the equation are plotted on a graph. Example: y = 3x + 7. The highest degree of a linear equation is always first. Lets understand it better with steps to solve linear equations and a few solved examples.
Steps to Solve a Linear Equation:
The aim of solving an equation is to find the value of the variable that makes the statements true. By following these steps one can solve the equation easily.
- By using the addition or subtraction property of equality the same number is added or subtracted on both sides of the equation.
- Using the multiplication property of equality multiplies the same number on both sides. 3. Using the division property of equality, divide every term by the same non-zero number on both sides.
- Other than these properties you can also factorize, expand, square, etc to solve the equation.
Solving a Linear Equation:
Before solving we must know what a linear equation is. A linear equation is a first-order equation. It may contain more than one variable. A linear equation with 2 variables is called a linear equation in two variables similarly for three variables it is called a linear equation in three variables and so on. Ex: x+3y+5 = 9. Here the degree of variables x and y is 1. The above-mentioned steps can be used to solve a linear equation.
Example: Solve x + 4 = 8 + x/3
Solution: To solve this, multiply the equation by 3 on both sided
3x+ 43 = 83 + x
3x + 12 = 24 + x
3x – x = 24 -12
2x = 12
Divide by 2 on both sides
2x/2 = 12/2
x = 6
You can solve the same equation by trial and error method also
Example: x + 4 = 8 + x/3
Solve by using 2 for x on both sides
2+ 4 = 8 + ⅔
6 8 + ⅔
Hence 2 is not the solution.
Now try by replacing x by 6.
6+4 = 8 + 6/3
10 = 8 + 2
10 = 10
LHS = RHS
Hence the value of x is 6.
Some Worked Examples on Solving an Equation
Here are a few examples on solving an equation. For more solved examples and information log on to the cuemath website.
1. Solve for x, if 4x2 – 8x + 4 = 0.
Solution: The given equation is quadratic. We can solve this by using the factoring method. We split the middle term such that the product of its components gives you the product of the first and the last term.
Here the middle term -8x is split into -4x and -4x. The product of (-4x) (- 4x) = 16x2 .It is the same as the product of the first term (4x2) and the last term (4) that is 4x2 4 = 16x2.
4x2 – 4x – 4x + 4 = 0
Now take out common from first two and the last two terms
4x (x – 1) – 4 (x – 1) = 0
Now we have (4x – 4) (x -1) = 0
Equating these equations to zero we get,
4x – 4 = 0 and x – 1 = 0
x = 4/4 = 1 or x = 1
- Solve = 8
Solution: By squaring on both sides we get
x/3 = 82 x/3 = 64
Multiply by 3 on both sides
(x/3)3 = 643
x = 192