Algebraic Equations

Algebraic equations are like a secret code. By using letters like x and y we can identify the relationship between numbers and unknown values. An algebraic equation’s generic form is P = O or P = Q, here P and Q are polynomials. The equation’s right and left sides will always be identical and balanced. An algebraic equation shows the relationship between two expressions by connecting them. The main aim of an algebraic equation is to find the value of an unidentified variable. 

For example, 5 + x = 8 is an algebraic equation. Here, we need to find the unknown value of ‘x’. Also, the value of ‘x’ will make both sides equal. 

The value of x = 3
Now, we can substitute the value of ‘x’ in the equation.
5 + 3 = 8.
 

History of Algebraic Equations
 

The history of algebraic equations dates back thousands of years. The term 'Algebra' was introduced in the 9th century by a Persian mathematician, Al-Khwarizmi. He is known as the father of algebra. His book named “Al-Kitab al-Mukhtasar fi Hisab al-Jabr wal-Muqabala” contributed to the systematic solving of equations.

Over time, algebra became widespread in Europe. In the 16th and 17th centuries, many mathematicians introduced new notations and methods of algebra. The foundation of modern algebra was established by mathematicians like Evariste Galois and Niels Henrik Abel. They made significant contributions to algebra.

In mathematics, algebra is a fundamental concept that helps to find solutions for various calculations. 



Key Concepts In Algebraic Equations
 

Various key concepts made up the foundation of algebraic equations. Expressions and equations of algebra contain variables, coefficients, constants, and mathematical operations. Knowing these key concepts will help us to solve real-life problems as well as mathematical calculations.

 

• Variables and Constants

Variables are the symbols in algebraic equations, that denote an unknown value. Variables are denoted as x, y, or any other letter. The value of constants will remain unchanged.  Constants have a fixed value. 

For example, \(2x + 4 = 10\) 
Here, ‘x’ is the unknown value.
4 and 10 are constants.

 

• Terms and Coefficients

A term in an algebraic equation is a component that is divided by addition and subtraction. A number that multiples the variable is known as a coefficient. Each term has a coefficient. 

For example, 
\(3x + 5 = 11\)
Here, 3x and 5 are the terms. 3 is the coefficient, which multiplies x. 

 

Students may easily get confused with algebraic expressions and algebraic equations. So let us look into the major differences between algebraic expressions and algebraic equations. 

Algebraic equations are one of the foundations of modern mathematics. Knowing this essential concept helps students understand and learn quickly about the algebraic principles. Learning the properties of algebraic equations is useful in finance, physics, engineering, and economics. 

• The arrangement of the numbers, which we add or multiply, does not make any change in the outcome. For example, an algebraic equation is \(a + b = b + a\) 
  That is, \(2 + 3 = 3 + 2 = 5\).
  
  Let us look into another illustration, \((a + b) + c = a + (b + c)\)
  Here the grouping of numbers does not affect the answer. Take a look at this,
  
  \((1 + 2) + 3 = 1 + (2 + 3) = 6\)

• Addition and subtraction are distributed across multiplication. For instance, 
  \(a × (b + c) = (a × b) + (a × c)\)
  
  Now we can apply value to these variables. 
  \(6 × (3 + 1) = (6 × 3) + (6 ×1) \)
  
  First, we can add \(6 × (3 + 1)\):
  \(6 × (3 + 1) = 6 × 4 = 24 \)
  Now, move on to the next set, i.e., \((6 × 3) + (6 × 1)\)
  \((6 × 3) + (6 × 1) = 18 + 6 = 24\)
  
  Both ways give the same answer. Now, we know that multiplication distributes over addition and subtraction. 

• In an equation, if we add zero or multiply by 1 does not make any change to the number. Let us look into this property in detail.
  
  For example, \(a + 0 = a \)
  That is: \(2 + 0 = 2\)
  
  Likewise, if we multiply a number by 1 in an equation, the digit will remain unchanged.
  \(b × 1 = b\)
  \(8 × 1 = 8\)

• We get identity values through the addition of the additive inverse or the multiplication by the reciprocal (multiplicative inverse). 
  
   
• An additive inverse is the opposite of a number. For example, the additive inverse of 5 is -5. 
  
   
• Multiplicative inverse refers to the number we used to multiply to get 1. For instance, the multiplicative inverse of 6 is \(\frac{1}{6}\).
  \(6 ×\frac{1}{6} = 1\)

The various types of algebraic equations are determined by the structure and the nature of the numbers involved in the formula. Algebraic equations show the equality of two algebraic expressions. Understanding types of algebraic equations helps students easily point out the right methods for solving calculations effectively. The main types of algebraic equations are listed below:

• Linear equations
• Quadratic equations
• Cubic equations
• Polynomial equations
• Rational equations 
• Radical equations

Let us take a closer look at each type of algebraic equation.

Linear Equations 

The highest degree of a variable in a linear equation is 1. Linear equations are also known as first-order equations. When we graph this equation, it forms a straight line. It is the simplest type of equation. The linear equations consist of constants like numbers and variables like x or y. 

The general form of linear equations is:
\(a_1x_1 + a_2x_2 +...+ a_nx_n + c = 0\)

In this, at least one of the coefficients is not zero. Take a look at this example,
\(ax + b = 0\)
\(ax + by + c = 0 \)

In the first equation, x is only one variable. In the second equation, x and y are the two variables. 

For instance, 
\(2x + 3 = 0 \)

This is a linear equation in one variable. Here, we have to find the value of ‘x’. To get the balance over two sides of the equation, we need to do the same thing on both sides.

Now, let us subtract 3 from both sides. 

Subtract 3 from \(2x + 3\), it gives \(0(+3 -3)\)

Subtract 3 from 0, it gives -3. 

 We get, \((2x + 3) - 3 = 0 - 3\) 

 So, \(2x = -3  \)

Here, 2x means two times x. Now, we have to divide both sides of the equation by 2.

\(\frac{2x}{2} = \frac{-3}{2}\)

\(x = \frac{-3}{2} = -1.5\)

Hence, the value of x is -1.5. 

\(2 × -1.5  + 3 = 0 \)
 

Quadratic Equations  

In a quadratic equation, the highest power of a variable is 2. This equation is also known as the second-degree equation.  When we graph this equation, it forms a U-shaped curve. The general form of a quadratic equation is: \(ax^2 + bx + c = 0\)

Here, x is the variable that is squared. The numbers are a, b, and c. The value of a is not equal to zero.          For example, \(3x^2 + 5x + 7 = 0\)

Now we can find the coefficients:

a = 3

b = 5

c = 7

To figure out the value of ‘x’, we need to apply the quadratic formula
 \(x = (-b ± \frac{√(b2 - 4ac)}{(2a)})\)

Let us substitute the values:
\(x = (-5 ± \frac{√(52 - 4 × 3 × 7)}{(2 × 3)})\)

Now, we can simplify each element:
\(5^2 = 25 \)
\(4 × 3 × 7 = 84\)
\(2 × 3 = 6\)

Now, we can subtract \(4ac - b^2\)

\(25 - 84 = -59\)

\(x = (-5 ± \frac{√(-59)}{6}) \)

√-59 is a negative number, so we use an imaginary number denoted by ‘i’.
\(√-59 = i√59\)

The final equation will be:
\(x = (-5 ±\frac{ i √-59}{ 6})\)

Cubic Equations  

The highest power of the variable in a cubic equation is 3. Cubic equations are also known as third-degree equations. These equations have real or complex numbers that depend on the coefficients. 

The general form of a cubic equation is:
\(ax^3 + bx^2 + cx + d = 0\)

Another example of cubic equations is:
\(2x^3 + 3x^2 - 5x + 4 = 0\)

a = 2 

b = 3 

c = -5

d = 4
 

Polynomial Equations  

A polynomial equation has a polynomial expression. The value of the polynomial is equal to zero. A polynomial equation is made up of variables, coefficients, and non-negative integer exponents. These equations have several terms with different powers, such as x² and x³. These variables can be added or subtracted together.

The general form of a polynomial equation is:
\(a_nx_n + a_{n-1}x{n-1} +…+ a_1x = a_0 = 0\)

To understand the polynomial equation better, look at these examples,
\(3x^3 - 5x^2 + 2x - 7 = 0\)
\(2x^2 - 3x + 1 = 0\)

Here, x is the variable, a_n, a_n-1, a₁, and a₀ are the coefficients. The degree of the polynomial is ‘n’ which        is a non-negative integer.

Rational Equations  

An equation containing fractions that have a variable in the numerator or denominator is known as a rational equation. By removing the fractions, these equations can be solved. Usually, the variable is in the denominator. These types of equations have two sides and have rational expression terms on each side. The general form of a rational equation is:

\(\frac{f(x)}{g(x)} = h(x)\)
Here, g(x) is the denominator.
f(x), g(x), and h(x) are the fractional variables. 

For a better understanding, take a close look at these examples of rational equations,
\(\frac{3}{x + 5} = 7 \)
\(\frac{2}{x + 3} = 5\)

When we solve rational equations, first, isolate the fraction. Next, multiply both sides by the denominator. After that, solve for the variable. 

Radical Equations
 

 In radical equations, a variable has a radical sign, such as a square root (√), cube root (³√), or any other root. The variable will be inside the root. The general form of the radical equation is:

\(n√f(x) = g(x)\)

The radical term in the equation is \(n√f(x)\).
f(x) is inside the root. 
g(x) can be a constant.

Some examples of radical equations are:
\(√(x + 4) = 6\). This is an example of a square root radical equation.
\(∛(x + 2) = 3\). This is an example of a cube root equation.
 

While an algebraic expression often uses identities, like \((a+b)^2\) for simplification, an algebraic equation requires specific formulas or methods to find the value(s) of the unknown variable(s) that satisfy the equality. These are typically organized by the degree (highest exponent) of the equation.

Solving Linear equations (Degree 1)

• Standard form: \(ax + b = c\)
• Slope-intercept : \(y = mx + b\)
• Point-Slope: \(y - y_1 = m(x - x_1)\)

Solving Quadratic equations (Degree 2)
 

• Standard form: \(ax^2 + bx + c = 0\)
• Quadratic Formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
• Discriminant: \(\Delta = b^2 - 4ac\)

Solving Cubic equations (Degree 3)
 

• General formula: \((ax^3 + bx^2 + cx + d = 0)\)

Solving an algebraic equation is the process of finding the unknown value (s) of the variable(s) that satisfy the equation. This value is known as the solution or root. The core principle in solving any equation is maintaining balance: whatever operation you perform on one side of the equal sign, you must perform the same operation on the other side.

The following steps are generally used to solve a linear equation:

Step 1: Simplify both sides (if necessary):
Before isolating the variable, simplify the expressions on both the left-hand side (LHS) and the right-hand side (RHS) by distributing or combining like terms. 
For example, using the distributive property, \(2(x+3)\) becomes \(2x+6\). And by the combining like terms method, \(5x + 2 - 3x\) becomes \(2x + 2\). 

Step 2: Isolate the variable term:
Use addition or subtraction to move all constant terms (numbers without a variable) to the side opposite the variable term.
For example, \(3x + 4 = 10\). Subtract from both sides:
\(3x + 4 - 4 = 10 - 4\)
\(3x = 6\)

Step 3: Solve for the variable:
Use multiplication or division (the inverse of the operation performed on the variable) to isolate the variable completely.
For example: \(3x = 6\)
Since x is multiplied by 3, divide both sides by 3.
\(\frac{3x}{3} = \frac{6}{3}\)
\(x=2\)

Step 4: Check the solution (verification):
Substitute the value you found for the variable back into the original equation to ensure both sides are equal, making the equation a true statement.
For example: Original equation: \(3x + 4 = 10\)
Substitute\( x = 2\): then, \(3(2) + 4 = 10\)
\(6 + 4 = 10\)
\(10 = 10\).
As 10 = 10 is true, the solution x = 2 is correct.

Algebraic equations are mathematical statements that show the relationship between two algebraic expressions. They are the fundamental concepts that pave the way for modern mathematical concepts. Knowing algebraic equations helps students to focus more on advanced math and its complicated calculations. Proper knowledge of these equations improves kids' problem-solving and logical reasoning skills. Knowing the concepts of algebra is essential for understanding calculus, statistics, and linear algebra. Algebraic equations help students to approach real-life problems such as budgeting, decision-making, and so on. 

If we properly understand the concepts of algebraic equations, the solving process of calculations becomes much easier. There are some tips and tricks that should help students to get mastery over algebraic equations. They are listed below:

• Knowing about the structure of equations helps kids to solve calculations efficiently.  An equation contains constants, coefficients, and variables. Also identifying which type of equation will enhance the overall performance. 
  
   
• Always remember to keep a balance for the equation. For example, \(2x + 4 = 10\). Here, \(x = 3\), so \((2 × 3) + 4 = 6 + 4 = 10\). If we apply any operations on one side, do the same operation on the other side. 
  
   
• If we are performing a quadratic equation, use the quadratic formula. The general form of a quadratic equation is \(ax^2 + bx + c = 0\). The formula \(x = (-b ± \frac{√b2 - 4ac} {2a})\) will help you solve operations easily. 
  
   
• When we solve rational equations, first we need to isolate the fraction, then multiply both sides by the denominator. 
  
   
• Whatever operation you do on one side of the equation, you must do the same on the other side to keep it balanced. For example, if you add 5 to the left side, you must also add 5 to the right side.
  
   
• Students often confuse the goal: simplifying an expression versus solving an equation. So, use clear, different instructions. For expressions, the instruction is "simplify" or "evaluate." The answer is an expression or a number. For equations, the instruction is "solve." The answer is a specific value for the variable.
  
   
• Ensure students can break down the structure of any equation before solving it. Use color-coding or visual boxes to group like terms before combining them, especially when variables appear on both sides of the equation.
  
   
• Parents and teachers can use simple stories, like “If a child has 5 chocolates and needs 10 to share with friends, how many more are needed?” This helps students see equations as everyday problem-solving tools, not abstract symbols.
  
   
• After solving, plug the value of the variable back into the original equation. This helps students develop independence and correctness.
  
   
• Try equation treasure hunts, card games, math puzzles, or simple riddles involving unknowns for students. Fun makes algebra less intimidating.

The applications of algebraic equations extend far beyond classrooms. They are important tools for modeling and solving problems across various fields. Some of its real-life applications are given below:
 

• Finance and economics: Equations are widely used to calculate expenditure, savings, and profit. They are essential for modeling market behavior, calculating simple and compound interest, and planning investment strategies.
  
   
• Engineering and architecture: Professionals use algebra to determine the dimensions, stability, and load-bearing capacity of structures. This includes calculating bridge stresses, sizing building beams, and designing machines.
  
   
• Physics and chemistry: Equations form the foundation of scientific laws. They are used to calculating variables such as speed, distance, and time in physics (d = rt), and to balance chemical reactions or determine concentrations in chemistry.
  
   
• Technology and computing: Algorithms and computer programs rely heavily on algebraic principles. Equations are used in everything from 3D rendering and video game physics to developing encryption methods and optimizing data flow.
  
   
• Everyday consumer decisions: Algebra helps in personal budgeting, comparison shopping, and calculating discounts or sales tax. For example, calculating the best per-unit price at the grocery store involves solving a simple division/rate equation.