About Equations in Math
What is an Equation?
Definition: A mathematical expression with an equal symbol is called an equation. Algebra is frequently used in equations. When you don’t know the exact number in a calculation, you’ll need to use algebra.
An equal symbol (=) must appear in an equation. In the equation the left-hand side (LHS) is equal to the right-hand side (RHS). The unknown in an algebraic equation is represented by the alphabet, such as x, y, z etc which are unknown as variables.
For example, x + 2 = 6 (x is an unknown variable).
Operations on equation
Addition, subtraction, multiplication, and division are examples of operations on equations. You are performing the addition operation on two numbers when you add them together. When you multiply two numbers together, you are also performing a multiplication operation. When you divide two numbers, you are also performing a division operation. You are performing the subtraction operation on two numbers when you subtract them together.
Addition property of an equality
Because addition is commutative and associative, the order in which a finite number of terms is added makes no difference. An equivalent equation is created by adding the same numbers on both sides of an equation.
- If p = q, then p + r = q + r (Adding ‘r’ on both sides)
For example, Solving equation y – 2 = – 4
y – 2 = – 4 (Writing the equation separately from question)
y – 2 + 2 = – 4 + 2 (using addition property of equality)
y = – 2 (Simplified)
Hence solution is y = – 2.
Check: y – 2 = – 4
– 2 – 2 = – 4 (substituted – 2 in the place of y)
– 4 = – 4
The answer is correct.
Subtraction property of an equality
Subtraction however is not commutative or associative in either way. An equivalent equation is created by subtracting the same numbers on both sides of an equation.
- If p = q, then p – r = q – r (Subtracting ‘r’ on both sides)
For example, Solving equation M + 2.3 = 4.6
M + 2.3 = 4.6 (Writing the equation separately from question)
M + 2.3 – 2.3 = 4.6 – 2.3 (using subtraction property of equality)
M = 2.3 (Simplified)
Hence solution is M = 2.3.
Check: M + 2.3 = 4.6
2.3 + 2.3 = 4.6 (substituted 2.3 in the place of M)
4.6 = 4.6
The answer is correct.
Multiplication property of an equality
Multiplication is distributive over addition and subtraction, as well as commutative and associative. An equivalent equation is created by multiplying the same numbers on both sides of an equation.
- If p = q, then p r = q r. (Multiplying ‘r’ on both sides)
For example, Solving equation \(\frac{Y}{3}=-7\)
\(\frac{Y}{3}=-7\) (Writing the equation separately from question)
\(\frac{Y}{3}\times (-3)=-7\times (-3)\) (using multiplication property of equality)
Y = – 21 (Simplified)
Hence solution is Y = – 21.
Check: \(\frac{Y}{3}=-7\)
\(\frac{-21}{3}=-7\) (substituted -21 in the place of Y)
– 7 = – 7
The answer is correct.
Division property of an equality
The dividend divided by the divisor generates the quotient of two numbers. Neither commutative nor associative division appears to exist. An equivalent equation is created by dividing the same numbers on both sides of an equation.
- If p = q, then \(\frac{p}{r}=\frac{q}{r}\) (Dividing with ‘r’ on both sides)
For example, Solving equation \(5\pi=\pi x\)
\(5\pi=\pi x\) (Writing the equation separately from question)
\(\frac{5\pi}{\pi}=\frac{\pi x}{\pi}\) (using division property of equality)
5 = x (Simplified)
Hence solution is x = 5.
Check: \(5\pi=\pi x\)
\(5\pi=\pi(5)\) (substituted 5 in the place of x)
\(5\pi=5\pi\)
The answer is correct.
What is the solution of an equation?
To solve an equation in mathematics, you must first find its solutions, which are the values (numbers, functions, sets, and so on) that satisfy the condition stated in the equation, which is usually made up of two expressions linked by an equals sign. One or more variables are designated as unknowns when looking for a solution. A solution is a set of values assigned to the unknown variables that ensure the equation’s equality. To put it another way, a solution is a value or a set of values (one for each unknown) that, when substituted for the unknown, makes the equation equal and true. The solution set of an equation is the collection of all of its solutions.
For example, What value of y is required for the equation y + 9\(\div\)2 = 12 to be true?
y + 9\(\div\)2 = 12 (Writing the equation separately from question)
y + 4.5 = 12 (Dividing 9 with 2)
y + 4.5 – 4.5 = 12 – 4.5 (using subtraction property of equality)
y = 7.5 (simplified)
Check: y + 9\(\div\)2 = 12
7.5 + 9\(\div\)2 = 12 (substituting 7.5 in the place of y)
7.5 + 4.5 = 12 (common properties of addition)
12 = 12
The answer is correct.
Solving multistep equations
The term ‘multi’ refers to more than two individuals or a large number of individuals. Because multi-step equations have more steps, they are more difficult to solve than one-step or two-step equations.
Step 1: By using the Distributive Property of Multiplication over Addition, you can get rid of any grouping symbols like square brackets and parentheses.
Step 2: If possible, combine like terms to simplify both sides of the equation.
Step 3: Decide where you would like the variable to be kept because this will help you to determine where to put the constant.
Step 4: Apply opposite operations to eliminate numbers or variables: addition and subtraction are opposite operations, but so are multiplication and division.
For example, Solving the equation 3(2 – 3x) + 2 = – 4
3(2 – 3x) + 2 = – 4 (writing the equation separately from question)
3(2) – 3(3x) + 2 = – 4 (using distributive property)
6 – 9x + 2 = – 4 (multiplying)
– 9x + 8 = – 4 (Combining like terms)
– 9x + 8 – 8 = – 4 – 8 (using subtraction property of equality)
– 9x = – 12 (simplified)
\(\frac{-9x}{-9}=\frac{-12}{-9}\) (using division property of equality)
x = 1.33 (simplified)
Check: 3(2 – 3x) + 2 = – 4
3(2 – 3(1.33)) + 2 = – 4 (Substituted 1.33 in the place of x)
3(2 – 4) + 2 = – 4 (using common properties of subtraction)
3(- 2) + 2 = – 4 (using common properties of multiplication)
– 6 + 2 = – 4 (using common properties of addition)
– 4 = – 4
The answer is correct.
Solved Examples of an Equation
Example 1: Solve w – 1 = 5
Solution:
w – 1 = 5 Write the equation
w – 1 + 1 = 5 + 1 Using addition property of equality
w = 6 Simplify
The solution is w = 6
Example 2: Solve \(\frac{2}{3}y\) = 5
Solution :
\(\frac{2}{3}y\) = 5 Write the equation
\(\left(\frac{2}{3}\times\frac{3}{2}\right)y=5\times \frac{3}{2}\) Using Multiplication property of equality
\(y=\frac{15}{2}\) Simplify
The solution is \(\frac{15}{2}\)
Example 3: A rectangle’s length is twice its breadth. Find the rectangle’s length and breadth if the perimeter is 30 meters.
Solution:
Consider the length of rectangle = 2x
The breadth of the rectangle = x
The perimeter of the rectangle = 30
According to the given data, 2(x + 2x) = 30
2(x + 2x) = 30 writing the equation
2(x) + 2(2x) = 30 using distributive property
2x + 4x = 30 multiplying
6x = 30 Combining like terms
\(\frac{6x}{6}=\frac{30}{6}\) using division property of equality
x = 5 simplified
Since, length of the rectangle = 2x = 2(5) (substituting 5 in the place of x)
Length of the rectangle = 10
breadth of the rectangle = x = 5
Check: 2(x + 2x) =30
2(5 + 10) = 30 (substituting x value as 5)
2(15) = 30 (using common properties of multiplication)
30 = 30
The answer is correct.
Example 4: Rock’s father is 3 times Rock’s age. After six years, Rock’s father will be twice his age. Calculate their current ages.
Solution:
Consider rock’s age = x
Rock’s father age = 3x
From the given question, after 6 years rock’s age = x + 6
The Rock’s father age = 3x + 6
According to the given question, 3x + 6 = 2(x + 6)
3x + 6 = 2(x + 6) (writing the equation separately from the question)
3x + 6 = 2(x) + 2(6) (Using distributive property)
3x + 6 = 2x + 12 (simplified)
3x + 6 – 6 = 2x + 12 – 6 (using subtraction property of equality)
3x = 2x + 6 (simplified)
3x – 2x = 6 (combining like terms)
x = 6 (simplified)
Rock’s age = x = 6 and that of his father’s age = 3x = 36 = 18.
Example 5: It costs $42 to buy 30 mangoes and 14 apples. Find the cost of the mango and the apple if the mango costs $1 more than the apple.
Solution:
The mango costs $1 more than the apples. Let’s consider the cost of the apple to be x
Then the cost of the mango = $1 + x
The cost of the 14 apples = 14x = 14x and the cost of 30 mangoes = 30(1 + x)
Total cost of the 30 mangoes and 14 apples = $42
Therefore, 30(1 + x) + 14x = 42
30(1 + x) + 14x = 42 (writing the equation separately)
30(1) + 30(x) + 14x = 42 (using distributive property)
30 + 30x + 14x = 42 (simplified)
30 + 44x = 42 (combining like terms)
30 – 30 + 44x = 42 – 30 (using subtraction property of equality)
44x = 12 (simplified)
\(\frac{44x}{44}=\frac{12}{44}\) (using division property of equality)
x = 0.27 (simplified)
And 14x = 140.27 = 3.78
Therefore, the cost of each apple is $0.27 and each mango is $3.78.
Check: 30(1 + x) + 14x = 42
30(1 + 0.27) + 3.78 = 42
30(1.27) + 3.78 = 42 (using common properties)
38.1 + 3.78 = 42
42 = 42
The answer is correct.
Like terms are those that have the same symbolic (variable) and the same exponents.
For example, x + 2x = 3x (here x and 2x are the like terms).
Unlike terms are those that have the same variable with different exponents or different variables with the same exponents.
For example, \(x^2+3x=1\) (here \(x^2\) and 3x are the unlike terms).
To ‘distribute’ something is to divide it or to give a share or part of it.
a\(\times\)(b+c)=(a\(\times\)b)+(a\(\times\)c)
Distributive states that “Multiplying the sum of two or more addends by a number produces the same outcome as multiplying for every addend individually by a number and then adding the products together.”