What is Algebraic Equation?
An algebraic equation is a mathematical equation that equates two algebraic expressions. It consists of coefficients, variables and constants as already introduced. In algebraic equations, the number of variables can be any finite number. The exponent of a variable can be positive, negative, or rational. We can write algebraic equations in more than one variable also. We will focus on equations with one and two variables in this article. For example:
- \( x + 4 = 0 \)
- \( x + xy + 9 = y^2 \)
- \( \sqrt[3]{x}+x=9 \)
Generally, we can represent an algebraic equation by P = 0 where P can be any polynomial.
Types of Algebraic Equations
The types of algebraic equations are –
- Linear Equations
- Quadratic Equations
- Cubic Equations
Linear Equations
A polynomial equation of degree 1 is called a linear equation.
The standard form of a linear equation in one variable is, \( ax+b=0 \), where \( a\neq 0 \)
The standard form of linear equations in two variables is, \( ax+by+c=0 \), where \( a\neq 0 \)
For example:
- \( 4x+5=0 \)
- \( x+y+1=0 \)
Quadratic Equations
A polynomial equation of degree 2 is called a quadratic equation.
The standard form of a quadratic equation in one variable is, \( ax^2+bx+c=0 \), in which \( a \neq 0 \)
For example:
- \( x^2-4x+6=0 \)
- \( x^2+6=0 \)
Cubic Equations
Polynomial equations of degree 3 are called cubic equations.
The standard form of cubic equations in one variable is, \( ax^3+bx^2+cx+d=0 \), in which \( a \neq 0 \)
For example:
- \( 2x^3-3x^2+4x-6=0 \)
- \( x^3-7x+9=0 \)
Solution of Algebraic Equations
The solution of algebraic equations are the values of variables that satisfy the equation. The value of a variable gives the same value when we substitute it in both the right and left hand side of the algebraic expression.
For example : Find the value of x that satisfies the equation \( 4x+5=3x+12 \).
Solution:
\( 4x+5=3x+12 \) Given Equation
\( 4x-3x=12-5 \) Combine like terms
\( x=7 \) Subtract
Substitute the value of \( x \) in the algebraic equation.
\( 4 \times 7 + 5 = 3 \times 7 + 12 \)
\( ~~~~~~~~~~~~33 = 33 \)
So, the left hand side and the right hand side are equal.
Therefore, \( x=7 \) is the solution for the given equation.
Solution of Algebraic Equations by Different Operations
We can solve the equations by using different mathematical operations such as addition, subtraction, multiplication, and division.
- Solution by Addition Method
Example: Solve \( x-4=3 \).
Solution:
\( x-4=3 \) Write the equation
\( ~+\underline{4}~~+\underline{4} \) Addition property of equality
\( x=7 \) Simplify
- Solution by Subtraction Method
Example: Solve \( x+4=3 \).
Solution:
\( x+4=3 \) Write the equation
\( ~-\underline{4}~~-\underline{4} \) Subtraction property of equality
\( x=-1 \) Simplify
- Solution by Multiplication Method
Example: Solve \( \frac{x}{4}=3 \).
Solution:
\( \frac{x}{4}=3 \) Write the equation
\( 4 \times \frac{x}{4}= 4 \times 3 \) Multiplication property of equality
\( x=12 \) Simplify
- Solution by Division Method
Example: Solve \( 4x=3 \).
Solution
\( 4x=3 \) Write the equation
\( 4 \times \frac{x}{4}=\frac{3}{4} \) Division property of equality
\( x=\frac{3}{4} \) Simplify
Rapid Recall
An algebraic equation is a mathematical equation that equates two algebraic expressions consisting of constants, coefficients and variables.
For example: \( 9x-12=5 \)
Solved Algebraic Equation Examples
Example 1: Solve \( 5(x-6)+18=8 \)
Solution:
\( 5(x-6)+18=8 \) Write the given equation
\( 5(x)-5(6)+18=8 \) Distributive property
\( 5x-30+18=8 \) Multiply
\( 5x-12=8 \) Combine like terms
\( 5x-12+12=8+12 \) Addition property of equality
\( 5x=20 \) Simplify
\( \frac{5x}{5} = \frac{20}{5} \) Division property of equality
\( x=4 \) Simplify
Hence, the solution is \( x=4 \).
Example 2: Solve \( 2(1-x)+7=0.5(4x-6) \)
Solution:
\( 2(1-x)+7=0.5(4x-6) \) Write the given equation
\( 2-2x+7=2x-3 \) Distributive property
\( 9-2x=2x-3 \) Combine like terms
\( 9-2x+2x=2x+2x-3 \) Addition property of equality
\( 9=4x-3 \) Simplify
\( 9+3=4x-3+3 \) Addition property of equality
\( 12=4x \) Simplify
\( \frac{12}{4} = \frac{4x}{4} \) Division property of equality
\( x=3 \) Simplify
Therefore, the solution is \( x=3 \).
Example 3: Sam has a total of \( \$ 90 \) in his account. He spent a certain amount on orange soda, and spent \( 5 \) more than twice the amount he spent on the soda, on pizza, and saved \( \$ 7 \). What is the amount he spent on the pizza?
Solution:
Sam has a total of \( \$ 90 \) and saves \( \$ 7 \) from it.
Let he spent \( x \) amount on soda.
So, he spent \( 2x+5 \) on pizza.
According to the above condition we can create an algebraic equation to find the amount spent on pizza,
\( x+2x+5+7=90 \) Writing the equation
\( 3x+12=90 \) Combine like terms
\( 3x+12-12=90-12 \) Subtraction property of equality
\( 3x=78 \) Simplify
\( \frac{3x}{3} = \frac{78}{3} \) Division property of equality
\( x=26 \) Simplify
Hence, Sam spent \( \$ 26 \) on the orange soda, and \( 2 \times 26+5=\$ 57 \) on pizza.
An equation is a mathematical statement that equates two algebraic expressions by using the equal sign between them.
Polynomial equations have constants, coefficients and variables. So, polynomial equations are algebraic expressions.
The various types of polynomial equations are linear equations, quadratic equations and cubic equations.
Yes, algebraic equations can have more than one variable. It can be one, two, three or more variables, and they are countably infinite.