Substitution Property

The substitution method is a way to solve a system of equations by replacing one variable with an expression from another equation, making it easier to find the values. 

Let's take an example.

Solve: \(x+y=5\\                        2x+3y=5\)

Explanation:

• Solve the first equation for one of its variables, for example: 
  \(y=5-x\)
   
• Substitute into the second equation. 
  \( 2x+3(5-x)=5\)
   
• Simplify and solve for x
  \( 2x+15-3x=5\\ -x+15=5\\ -x=-10\\ x = 10\)
   
• Back-substitute to find y 
  \(y=5-x\\ y=5-10\\ = -5\)

Answer: \(y=-5 \ and \ x = 10\)
 

The substitution property of equality states that if two quantities are equal, one can be substituted for the other in any expression or equation.

For example, if a = b, then we can replace 'a' with 'b' in any expression, and the value of the expression won’t change.

In example, if a + 2 = 0, and a = b, we can substitute a with b, and the expression becomes b + 2 = 0.    

Let's understand it better with a problem.

Expression to evaluate: \(x^2-3x+8\)
Given: x = 1

Using the substitution property, we replace x by 1
\(1^2-3(1)+8=1-3+8=6\)

So, the expression evaluates to 6 when x = 1.

Parent Tip: Encourage your child to first practice substitution in linear equation in one variable and then move to quadratic equations.

To solve an equation using substitution property, follow the steps mentioned below:

1. Isolate a variable from one equation where the coefficient is 1 or -1.
    
2. Substitute that expression into the other equation, which creates a single-variable equation.
    
3. Solve the resulting equation for that one variable.
    
4. Back-substitute the found value into the expression from step 1 to find the other variable.
    
5. Check your answers by plugging both values into the original equation. If both sides are equal, your solution is correct. 

The following flow chart is the step-by-step breakdown of solving a system of equations using substitution method.

Let's practice this using a problem.

Practice Problem: Solve: \(x + y = 20\\x-y=10\)

Explanation:  The given equations:

• x + y = 20 — (1)
• x - y = 10 — (2)

1. Isolating equation 2: x = y + 10
    
2. Substituting to find the value of x:
   \(x + y = 20\\ (y + 10) + y = 20\\ 2y = 20 - 10\\ 2y = 10\\ y = 5\)
    
3. Substituting the value of y in equation 2: 
   \(x - y = 10\\ x - 5 = 10\\ x = 10 + 5 \\ x = 15\)

The substitution method involves solving one equation for a variable. For example, rewriting x - y = 2, as x = y + 2, and then plugging that into the other equation. It's intuitive and works best when one variable is easily isolated.

The elimination method multiplies one or both equations by suitable numbers to make the coefficients equal. Then add or subtracts them to cancel out one variable. It's often faster and avoids fractions when coefficients are already equal or opposites. 

Let's take a system of equation and solve with both methods to understand the difference.

To help you grasp the concept of substitution property and solve problems more effectively, here are some tips and tricks.

1. Remember, put the value of a variable derived from one equation into the second equation.
2. Correctly perform arithmetic operations, and double check for mistakes.
3. Substitute the values of variable in the original equation to verify your answers.
4. Always use parenthesis for substitution to avoid miscalculations.
5. If the equation has parenthesis, solve it first.

Parent Tip: To explain substitution property to your child, you can use real-world examples. Like, nickname and the legal name of your child belongs to the same person. You can use both in conversations. Encourage your child to practice problems.

Real-life applications are important in chemistry, physics, and economics, and are used in many ways. Here are examples of real-life applications mentioned below

1. Budgeting: Students can use the substitution method to manage their budget and to plan expenses like food, books, or transportation.
   
   For example, you receive $20 as allowance and earn $40 from a part-time job, you substitute these values into an equation to find your total income as $60.
   
    
2. Converting Unis: The substitution method is used to convert measurement units.
   
   For example, 1 meter = 100 centimeters, then 2 meters = 2 × 100 = 200 centimeters. 
   
    
3. Balancing chemical equations: In chemistry, we can balance and simplify chemical equations, using substitution method. If one coefficient is unknown, it can be expressed in terms of another to make sure both sides are equal.
   
   For example, in a reaction like: xA + yB = zC (where A, B, and C are chemical substances and x, y, and z are their coefficients), we use substitution to relate the coefficients based on the number of atoms needed to find the values.     
   
    
4. Calculating Time: To calculate the travel time, we use the substitution method, which helps students plan their day.
   
   For example, if a student bikes to school at 10 mph, to find the time, we use the formula, distance = speed × time.
   
    
5. Temperature Conversion: We can use substitution method to convert temperature from one unit to others. 
   
   For example, since, K = 273.15 + °C, that implies 2°C can be written as, K = 273.15 + 2°C = 275.15K by using substitution.