Solutions of Linear Equations

A linear equation is an algebraic equation in which the highest power of the variable is 1. The values increase or decrease evenly, forming a straight-line graph. Coefficients multiply variables, while constants remain unchanged.
 

The solution to a system of linear equations can have three types of solutions, which are given below:

• Unique Solution
• No Solution
• Infinite Solution

1. Unique Solution 
   
   A linear equation in one variable has a unique solution, where the LHS is equal to the RHS. A linear equation in two variables is represented on a graph by a straight line, and its solution is expressed as an ordered pair (x, y).
   
    
2. No Solution 
   
   In a linear equation, a system with no solution is also known as an inconsistent solution. The non-solution happens when two lines never intersect. In a system of two linear equations, if the slopes are the same, but the intercepts are different, the lines are parallel, and there will be no solution.
   
    
3. Infinite Solutions
   
   The infinity solution occurs when the equations are on the same line. If both equations describe the same line, the lines overlap completely, and these are called coincident lines.

A linear equation contains only one variable in the expression. To solve a linear equation in one variable, keep the variable on one side and the constant on the other side. Then simplify the equation.

Example, 3x - 7 = 5

Solution:

• Keep the constant on one side and the variable on the other side, 
  \(3x -7 = 5\\ 3x = 5 + 7\\ 3x = 12\\ x = 12/3\\ x = 4\)

The value of x is 4.

How to Solve a Linear Equation in Two Variables?

The linear equation in two variables has two unknowns, usually x and y, and can be written as ax + by = c. Let's understand this using an example.

Example, x + y = 6, find the value of x and y.

Solution:
 

• We take the first two natural numbers instead of x:
  \(x + y = 6\)
   

1. x = 0
   \( y = 6 -x\\ y = 6 -0\\ y = 6\)
   
   x = 0
   y = 6
    
2. x = 2
   \(x + y = 6\\ y = 6 -1\\ y = 5\)
   
   x = 1
   y = 5

The methods used to solve linear equations are used in the solution of linear equations in two variables. There are several methods:

1. Substitution
2. Elimination
3. Graphical method
4. Cross multiplication method

• Substitution Method:
  
  The substitution method is a way to solve linear equations by expressing one variable in terms of the other. Then substitute the expression into the second equation. This leaves an equation with only one variable, which helps to solve the equation and find the value.

Example:
x + y = 12   (1)
x = 4           (2)

Solution:

Substitute equation 2 in equation 1
x + y = 12
4  + y = 12
y = 12 -4
y = 8

So, x = 4 and y = 8.

• Elimination Method 
  
  The elimination method is a method for solving linear equations. In this method, one of the variables is multiplied by a constant so that either the x term or the y term cancels out when the equations are added or subtracted, making it possible to find the value of the remaining variable. 

Example:
x + y = 12  (1)
x - y = 4    (2)
Solution: 
 

1. Add both equations
   \(x + y = 12  \\ x - y = 4   \\ (x + y) + (x -y) = 12  + 4\\ 2x = 16\)
   
   y is eliminated because they both have a different sign
   \(2x = 16\\ x = 16/2\\ x = 8\)
    
2. Substitute x = 8 in equation 1
   \( 8 + y = 12\\ y = 12 -8\\ y = 4\)

The values of x and y are 8 and 4 

• Graphical method
  
  The graphical method is used to solve the linear equation, graphing the equations by finding the values of x and y in the coordinate system. After finding the value, find the intersection point of these two lines. The value of the point gives the solution for these linear equations.

Let's practice.
Problem: Find the intersection plot for lines:
 y = x + 1
 y = −x + 3
Solution

Slope intersection form: y = mx + c
m is the slope
c is the intercept of y

• Step 1: Pick the values for x and calculate the y in equation 1
  y = x + 1
   

1. -1 in x
   y = -1 + 1 =0
   y = 0
    
2. When x = 0,
   y = 0 + 1
   y = 1
    
3. 1 in value of x
   y = 1 + 1
   y = 2
    
4. 2 in value of x
   y = 2 + 1
   y = 3

(x, y) = (-1,0), (0,1), (1,2), (2,3)

• Step 2: Pick the values for x and calculate y in equation 2
   y = −x + 3

1. When x = 0
   y = 0 + 3 = 3
   y = 3
    
2. 1 in value of x 
   y = -1 + 3 = 2
   y = 2
    
3. 2 in value of x
   y = -2 + 3 = 1
   y = 1
    
4. 3 in value of x
   y = -3 + 3 = 0
   y = 0

(x, y) = (0,3), (1,2), (2,1), (3,0)

• Step 3: Draw a straight line through the points.
  
  
  Both lines intersect at the point: (1,2)

• Cross multiplication method
  
  The cross multiplication method is used to solve linear equations by taking the coefficients of x, y, and the constant terms from the given equations.
  \(a_1x + b_1y + c_1 = 0\\ a_2x + b_2y + c_2 = 0\)
  
  The formation of the cross method is given below:

Let's see the example using the cross multiplication method
2x + 3y = 17
3x −2y = 6

Solution:
 

1. Write in the form like
   \(a_1x + b_1y + c_1 = 0\\ a_2x + b_2y + c_2 = 0\)
   
   2x + 3y - 17= 0 (a₁ = 2, b₁ = 3, c₁ = -17)
   3x −2y - 6 = 0 (a₂ = 3, b₂ = 2, c₂ = -6)
    
2. Substitute the values in the formula:
   \(\frac{x}{ (3 × -6 - 2 ×-17)} = \frac{y }{(-17 × 3 - (-6) ×2)} = \frac{1 }{(2 × 2 - 3 ×3)}\)
   
   \(\frac{x}{ (-18 +34)} = \frac{y }{(-51 + 12)} = \frac{1 }{(4 - 9)}\)
   
   \(\frac{x}{ 16} = \frac{y }{-39} = \frac{1 }{-5}\)
    
3. Now compare
   \(\frac{x}{ 16} = \frac{1 }{-5} \implies x = {16 \over -5}\)
   
   \( \frac{y }{-39} = \frac{1 }{-5} \implies y = {39 \over 5}\)

Study Strategy: If the system of equations has larger numbers as coefficients and constant, use cross-multiplication method.

Now, let's see some essential tips and tricks that will help you master solutions of linear equations

1. Always substitute the value of one variable obtained from an equation into the another equation.
    
2. Simplify the expressions if needed for easy calculation.
    
3. To verify your answers, put the values in the original equations. If it satisfies the equation, the answer is correct.
    
4. Always make the changes on both sides of the equations. 
    
5. Don't mix up signs.

Parent Tip:
 

• You can use real life situations to form linear equations and ask your child to solve those.
   
• Encourage your child to ask for help if they find it difficult.
   
• You can use calculators for evaluating linear equations and visual aids for better understanding.

Linear equations are not only for academia, but it is also used in our day-to-day lives. Here are some real-life applications where linear equations are required.

1. Budgeting and Financial Planning: Linear equations are commonly used to manage personal or household budgets. They can help to understand how the income is divided between savings and expenses, such as rent, food, and bills. It helps to control finances.
   
    
2. Shopping and Billing: Linear equations can be used to calculate the total cost, apply discounts, and determine how many items can be purchased within a set budget.
   
    
3. Travel and Transportation: Linear equations are useful for planning trips. It can help to calculate how long it will take to reach a destination based on speed and distance.
   
    
4. Business and Profit Analysis: Businesses use linear equations to understand their costs and profits. They calculate how many products need to be sold to break even or earn a certain amount of profit. This can help to grow the business steadily.
   
    
5. Cooking and Recipe Adjustments: In cooking, the linear equations are used to adjust the recipe quantities proportionally when increasing or decreasing the servings.