Equations
Equations
In mathematics, an equation is a statement that connects two expressions with the equals (=) sign to express their equality.
Example: Suppose we have a seesaw in the park. The seesaw will be balanced if we put equal weight on both ends of the seesaw, and if the weight is not equal, then it will not be balanced.
Difference between equation and expression
Equation | Expression |
|---|---|
A variable or number operation with an equal sign is called an equation. | An equation is a mathematical operation that involves variables or numbers but does not include an equal sign. |
Example: 2a + 7 = 1 | Example: 3x + 2y + 7 |
Solving equations by using different operations such as addition and subtraction
When two equations have the same solution, they are said to be equivalent. Equivalent equations can be created using the addition and subtraction properties of equality.
Properties
Property of addition for equality
The two sides of an equation remain equal when the same number is added to each side.
If a = b then a + c = b + c
Property of subtraction for equality
The two sides of an equation remain equal when the same number is subtracted from each side.
If a = b then a – c = b – c
Examples
Example 1: Solve the following equations using addition and subtraction operations.
(a) 15 = x – 3
(b) a + 3 = 8
Solution:
(a)
We have: 15 = x – 3
15 = x – 3
15 +3 = x – 3 + 3 Using the addition property of equality
18 = x
The solution is x = 18.
(b)
We have: a + 3 = 8
a + 3 = 8
a + 3 -3 = 8 – 3 Using the subtraction property of equality
a = 5
The solution is a = 5.
Example 2: The cost of a remote-control plane is $275.95. Your father gives you $50 to help you purchase the remote-control plane in the picture. You have $25.50 left after purchasing the plane. Before your father gave you $50, how much money did you have?
Solution: Write an equation using the word sentence of the given question:
Left Amount ( $ ) = Initial Amount ( $ ) + Father’s Money ( $ ) – Cost of the Plane ( $ )
Let the initial amount be i.
Form an equation.
25.50 = i + 50 – 275.95
25.50 + 275.95 = i + 50 – 275.95 + 275.95 Using the addition property of equality
301.45 = i + 50
301.45 – 50 = i + 50 – 50 Using the subtraction property of equality
251.45 = i
The total amount of money you have is $251.45.
The steps for using the subtraction and addition properties to solve equations are:
Step 1: Replace the variable in the equation with the number.
Step 2: Simplify both sides of the equation’s expressions.
Step 3: Check to see if the resulting equation is correct.
The number is a solution if it is true.
One-step equations can be solved in four ways: adding, subtracting, multiplying, and dividing.
A one-step algebraic equation is one that can be solved in a single step. When you get the variable on one side of the equal sign by itself, with no numbers in front of it, you’ve solved the equation.