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Substitution Property

Substitution property in algebra is a property that helps in substituting the value of a quantity/variable into an algebraic expression to solve a given mathematical problem. We have the substitution property of equality which gives the value of a variable and helps in finding the value of an algebraic expression by substituting the value of the variable. Another substitution property that we use in geometry gives the value of a geometrical aspect such as the measure of angle, length of a line segment, etc. whose value can be substituted in an expression to find the unknown.

In this article, we will explore the different types of substitution property and their applications with the help of a few solved examples for a better understanding of the concept.

1.	What is Substitution Property?
2.	Substitution Property of Equality
3.	Direct Substitution Property
4.	Substitution Property Geometry
5.	FAQs on Substitution Property

What is Substitution Property?

The substitution property is a concept in algebra that is used to substitute the value of a given variable or a quantity into an expression to find the value of the unknown. In simple words, we can say that the substitution property is used to replace the value of a quantity in an equation or an expression to solve a mathematical problem. We will discuss the meaning of the substitution property of equality, the substitution property in geometry, and the direct substitution property in limits in the following sections.

Substitution Property of Equality

The substitution property of equality states that 'If a variable x is equal to another variable y, then x can be substituted in place of y in any equation/expression and y can be substituted in place of x in any equation/expression.' This property is very useful in finding the value of an algebraic expression. The substitution property of equality helps in plugging in the values of the variables whose values are given to find the value of the unknown variables. Let us consider an example to understand the substitution property.

substitution property

Example: Find the value of the algebraic expression x² - 3x + 8 if x = 1.

Solution: We are given x = 1, so using the substitution property, we will substitute x with 1 in the given expression to find its value.

x² - 3x + 8 if x = 1

= 1² - 3(1) + 8

= 1 - 3 + 8

= -2 + 8

= 6

So, the value of the given algebraic expression x² - 3x + 8 is 6 when x = 1.

Direct Substitution Property

The direct substitution property is used to find the limit of a function by determining the value of the function at that point. As the name suggests, to find the limit of a function, we directly substitute the value of the variable into the function. In simple words, we can say that using the direct substitution property, we find the limits of functions at specific points by directly plugging the points into the function. Let us consider an example to understand the application of this property.

Example: Find the limit of the function f(x) = x³ - 4x² + x - 3 when x tends to -2.

Solution: To find the limit of the function at x = -2, we will use the direct substitution property. We will substitute the value of x by -2 into the expression lim_x→-2(x³ - 4x² + x - 3). So we have

lim_x→-2(x³ - 4x² + x - 3)

= (-2)³ - 4(-2)² + (-2) - 3

= -8 - 4×4 - 2 - 3

= -8-16-5

= -29

So, the limit of f(x) = x³ - 4x² + x - 3 when x tends to -2 using the direct substitution property is -29.

Substitution Property Geometry

The substitution property in geometry states that if a geometric figure such as a line segment, triangle, angle, etc. is congruent to another geometric figure, and a geometric problem is given to determine the value of the unknown, then we can substitute the value of one with another and solve the problem. Let us solve an example to understand the application of the substitution property in geometry.

Example: A line segment of length 5 cm is congruent to the side of a square. Find the perimeter of the square.

Solution: Since the line segment is congruent to the side of the square, using the substitution property in geometry, the length of the side of the square is 5 cm. Now, the perimeter of the square is given by,

Perimeter = 4 × 5 cm

= 20 cm.

Important Notes on Substitution Property

The substitution property is used to replace the value of a quantity in an equation or an expression to solve a mathematical problem.
We have different types of substitution properties in different mathematical areas such as substitution property of equality, substitution property in geometry, and direct substitution property.

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Substitution Property Examples

Example 1: Using the substitution property of equality, find the value of the algebraic expression g(x) = sin x + cos x if x = 45°.

Solution: To find the value of g(x) = sin x + cos x if x = 45°, we will replace x with 45° in the expression. So we have

g(45°) = sin 45° + cos 45°

= 1/√2 + 1/√2

= 2/√2

= √2

Answer: The value of the algebraic expression g(x) = sin x + cos x if x = 45° is √2.
Example 2: Angle A in triangle ABC measures 50°. Another angle B in the triangle is congruent to an angle of measure 30°. Find the measure of the third angle C.

Solution: Using the angle sum property of a triangle, we have

∠A + ∠B + ∠C = 180°

Since ∠B is congruent to another angle of measure 30°, ∠B = 30° using the substitution property in geometry. So, we have

∠A + ∠B + ∠C = 180°

⇒ 50° + 30° + ∠C = 180°

⇒ 80° + ∠C = 180°

⇒ ∠C = 180° - 80°

= 100°

Answer: The measure of angle C is 100°.
Example 3: Find the limit of the function f(x) = x² - 5 using the direct substitution property when x tends to 0.

Solution: To find the limit of f(x) = x² - 5 using the direct substitution property, we will substitute the value if x = 0 into the expression lim_x→-2(x³ - 4x² + x - 3).

lim_x→0(x² - 5)

= 0² - 5

= 0 - 5

= -5

Answer: The limit of the function f(x) = x² - 5 when x tends to 0 is -5.

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Substitution Property Questions

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FAQs on Substitution Property

What is Substitution Property in Maths?

The substitution property is used to replace the value of a quantity in an equation or an expression to solve a mathematical problem. We have different types of substitution properties in different mathematical areas such as:

substitution property of equality,
substitution property in geometry, and
direct substitution property.

What is the Difference Between Substitution Property and Transitive Property?

The substitution property is used to substitute the value of one variable/quantity in any expression or equation. On the other hand, the transitive property is used when when one quantity is equal to the second and the second is equal to the third which implies that the first is also equal to the third. In simple words, the transitive property is used when two quantities are equal to the same thing.

What is the Substitution Property in Geometry?

What Does the Substitution Property of Equality State?

How Does the Substitution Property Work?

The substitution property of equality helps in plugging in the values of the variables whose values are given to find the value of the unknown variables.

What is Direct Substitution Property in Limits?

We can say that using the direct substitution property, we find the limits of functions at specific points by directly plugging the points into the function.

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