Solving Equations

Solving equations is the process of finding the unknown variable that makes both sides of the equation equal. An equation is a mathematical statement where two expressions, involving a variable, are equal.

In these equations, the LHS and RHS can be interchanged, as both sides represent the same value.

There are different ways to solve an equation depending on its type, such as linear, quadratic, rational, or radical equations.

Solving an equation includes using mathematical operations to isolate the variable to find the value of the unknown variable. This value is found by isolating the variable using mathematical operations. Let’s look at the steps to solve an equation.

• Addition property of equality: If we add the same number to both sides, the equality is maintained.
  
  If \(a = b\), then \(a + c = b + c\)
   
• Subtraction property of equality: If we subtract the same number from both sides, the equation remains balanced.
  
  If \(a = b\), then \(a - c = b - c\)
   
• Multiplication property of equality: If we multiply both sides by the same number, the equality is not affected.
  
  If \(a = b\), then \(ac = bc\), for any number c
   
• Division property of equality: If we divide both sides by the same number (except zero), the equality is maintained.
  
  If \(a = b\), then \(\frac ac = \frac bc\) (where \(c ≠ 0\)).
   

We isolate the variable on one side of the equation after completing these steps.

The linear equation in one variable is expressed in the form \(ax + b = 0\), where a and b are real numbers. To solve such equations, follow these steps:
 

• Remove parentheses by applying the distributive property, if needed.
   
• To simplify the equation, we combine like terms.
   
• We eliminate fractional terms from equations by multiplying both sides by the least common denominator (LCD).
   
• If the equation has decimals, multiply both sides by the appropriate power of 10 to convert them into whole numbers.
   
• Apply the addition or subtraction property of equality to bring variable terms to one side and constants to the other.
   
• We use the multiplication or division property of equality to make the coefficient of the variable equal to 1.
   
• Isolate the variable to find the solution.
   

For example:

Solve the equation: \(3(x + 4) = 24 + x\)

Apply the distributive property on the LHS:

\(→ 3x + 12 = 24 + x\)

Group the like terms to one side :

\(→ 3x - x = 24 –12\)

Simplify both sides:

\(→ 2x = 12\)

To isolate x, we divide both sides by 2:

\(→ x = 6\)

Solution: \(x = 6\)

Using the trial-and-error method, we test different values of the variable until we find the one that satisfies the equation.

For example:

Consider the equation \(5x = 35\).

Look for a number that, multiplied by 5, gives 35

We determine \(x = 7\) since \(5 × 7 = 35\).

This method works well for simple equations, but for more complicated ones, it can become challenging and time-consuming.

Some equations can have more than one solution. This is often the case with quadratic equations, which are equations of degree two. The zeroes of a quadratic polynomial are the values that satisfy the equation.

Example:

\((x + 3)(x + 2) = 0\)

This is a quadratic equation that can be solved by writing each factor equal to zero:

\(x + 3 = 0 ⟹ x = -3\)

\(x + 2 = 0 ⟹ x = -2\)

So, the solutions are \(x = -3\) and \(x = -2\).

A quadratic equation is generally written in the form:

\(ax² + bx + c = 0\)

When a quadratic equation is solved, up to two roots are obtained: α and β.

We can solve a quadratic equation in different steps:
 

• Using the completing the square method 
   
• Using the factorization method
   
• Using the formula method

Completing the square method systematically solves a quadratic equation by applying the algebraic identity:

\((a + b)^2 = a^2 + 2ab + b^2\)
 

• We first need to express the equation in standard form:
  
  \(ax^2 + bx + c = 0.\)
   
• Divide the entire equation by ‘a’.
   
• Shift the constant term to one side of the equation.
   
• Add the square of half the coefficient of x to both sides.
   
• Complete the left-hand side as a perfect square.
   
• Take the square root of both sides.
   
• Determine the value of x.

A quadratic equation can be solved using the factorization method as discussed below:
 

• First, express the equation in standard form: \(ax² + bx + c = 0\)
   
• Split the middle term:
  
  Break the middle term, bx, into two terms such that: 
  
  Their sum equals b
  
  Their product equals \(a × c\)
   

For example:

Solve: \(2x² + 19x + 30 = 0\)

Find two numbers that add up to 19 and multiply to 60 (2 × 30)

→ 4 and 15

Now rewrite the equation:

\(2x² + 4x + 15x + 30 = 0\)

Group and factor:

\(2x(x + 2) + 15(x + 2) = 0\)

Take the common factor:

\((x + 2)(2x + 15) = 0\)

Now solve each factor:

\(x + 2 = 0 ⇒ x = -2\)

\(2x + 15 = 0 ⇒ x = \frac {-15}{2}\)

When the equation is of the form \(ax² + bx + c = 0\), we use the quadratic formula:

\(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)

To find the solution, we substitute the values of a, b, and c into the formula.

For example:

Solve: \(9x² - 12x + 4 = 0\)

Here, \(a = 9, b = -12, c = 4\)

Apply the quadratic formula:

\(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)

\(x = \frac{-(-12) \pm \sqrt{(-12)^2 - 4 \times 9 \times 4}}{2 \times 9} \)

\(x = \frac{12 \pm \sqrt{144 - 144}}{18} \)

\(x = \frac{12 \pm \sqrt{0}}{18} \)

\(x = \frac{12}{18} \)

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

Solution: \(x = \frac{2}{3} \)

A rational equation has at least one variable in the denominator. To solve it:
 

• We determine a common denominator or cross-multiply.
   
• Solve the resulting equation.
   

For example:

Solve: \(\frac{2x}{x + 4} = \frac{4}{5} \)

Cross-multiplying gives:

\(5 × 2x = 4(x + 3)\)

\(10x = 4x + 12\)

\(10x - 4x = 12\)

\(6x = 12\)

\(x = 2\)

A radical equation is an equation in which the variable is enclosed in a root. To solve it:
 

• Isolate the radical expression.
   
• Remove the radical by squaring both sides.
   
• Solve the equation obtained.

Example:

Solve: \(\sqrt{(2x - 3)} = 5\)

Square both sides:

\((\sqrt{(2x - 3)})² = 5²\)

\(2x - 3 = 25\)

\(2x = 28\)

\(x = 14\)

Here are some of the tips and tricks that will be helpful for the learners to master solving equations:
 

1. Before solving the equations, read and understand the equations carefully. Identify the unknown and identify the constants and coefficients.
    
2. Always remember to keep the equation balanced. Whatever we do on one side, must be done on the other side too. Add or subtract both sides using the same number. Multiply or divide by the same number on both the sides. 
    
3. Always try to perform simplification step by step. Combine like terms and remove brackets using the distributive property. Simplify the fractions if necessary.
    
4. Use inverse operations carefully. Do the opposite operation to isolate the unknown. Addition and subtraction; multiplication and division; powers and roots. 
    
5. Always remember to check your solutions. Substitute your answer back into the original equation to ensure that it works. 

Solving equations is a fundamental concept in mathematics, and we use it in different fields. Let’s now learn about their importance in real life. Here are a few real-life applications of solving equations.
 

• Shopping discounts: We can use equations to find the final price of an item after a discount. For example, if an item costs $2000 and is offered at a 50% discount, the final price can be calculated using the equation:
  
  Final price = original price - (discount price × original price)
  x = 2000 – 0.50 × 2000
  x = 2000 – 1000 = 1000
  So, the final price is $1000.
   
• Business:  The profit earned can be calculated using an equation. For example, if earnings = $2,00,000 and expenses = $60,000, the profit can be determined by solving:
  
  x = 200000 – 60000, which is $140000 (profit).
   
• Travel calculations: This concept can also be used to estimate the time needed to complete a journey. For example: if the total distance is 120 km and the vehicle travels at a speed of 60 km/h, the time can be calculated using the equation:
  
  Time = Distance/ Speed 
           = 120/60
           = 2 hours (estimated time).
    
• Cooking and recipes: We can use equations to adjust recipes proportionally. These equations can help scale ingredients accurately.
         
• Daily life problem-solving: These equations can be helpful in sharing food equally whenever necessary. They help a lot in planning different schedules and calculating travel expenses. These often require forming and solving equations.