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. 

For example: x+y=5
                       2x+3y=5

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

For example 
x = 1
Expression to evaluate: x2-3x+8
Using the substitution property, we replace x by 1
12-3(1)+8=1-3+8=6
So, the expression evaluates to 6 when x = 1.
 

1. Isolate one variable in 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. 

Solving: x+y=20
x-y=10

The given equations: x+y=20 — (1)
x-y=10 — (2)

Isolating equation 2: x = y + 10

Substituting to find the value of x:
x + y = 20
(y + 10) + y = 20
2y = 20 - 10
2y = 10
y = 5

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 it 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 adds or subtracts them to cancel out one variable. It's often faster and avoids fractions when coefficients are already equal or opposites. 
 

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

• Students can use the substitution method to manage their budget. For example, if they receive $50 as pocket money and earn $300 from a part-time job, they can substitute these values into an equation to represent their total income as $350. This total can be used to plan expenses like food, books, or transportation. 

• The substitution method is used to convert the units. For example, 1 meter = 100 centimeters, then 2 meters = 2 × 100 = 200 centimeters. 

• In chemistry, to balance and simplify chemical equations, we use the 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.     

• 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.