The word substitute means to replace one thing with another. So, what does substitute mean in math? In math as well, substitute means to replace something with another such as replacing a variable with a value, replacing a complex equation with a variable etc. Let us how the substitution is used in various operations in mathematics.
Let us recall that a polynomial is an algebraic expression that is formed through a combination of constants and variables. For example, 6x + 4 is a polynomial having two terms, 6x and 4. Similarly, 3×2 + 5x + 7 is a polynomial having 3 terms, 3×2, 5x and 7. When a polynomial is equated to zero of any other value, it forms an equation. For example, if we equate the polynomial 6x + 4 = 0, an equation is formed. Such equations can be solved for values of the given variable, in this case, x. There are many methods to solve such equations, one of which is the substitution method. Let us take an example
Example
For what value of x, would the value of the following polynomial be 8?
2x + 4
Solution
We have been given the polynomial, 2x + 4. We need to find the value of x for which the result of this polynomial is 8. Using the substitution method, let first substitute x = 1 in the given polynomial. We will have,
2x + 4 = 2 ( 1 ) + 4 = 2 + 4 = 6 which is not the desired result. . . . . . . . . . . . . . . . . . ( 1 )
Next, we shall substitute x = 2 in the given polynomial. We will have,
2x + 4 = 2 ( 2 ) + 4 = 4 + 4 = 8 which is the desired result. . . . . . . . . . . . . . . . . . . ( 2 )
From ( 1 ) and ( 2 ), we can see that if we substitute x = 2 in the given polynomial, we get 8 as the answer. Hence, the solution for the given polynomial is x = 2.
Such a method of substituting different values in the variable to check the result is also known as the hit and trial method.
Let us now learn about the substitution method to solve equations.
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We know that an equation has components, left hand side (LHS), right hand side (RHS) and an equal sign ( = ) that separates LHS and RHS. For example, 5x + 8 = 23 is a simple linear equation. Now, how can we use the substitution method to solve simple linear equations? Let u find out.
We know that a single linear equation in one variable is of the form, ax + b = 0, where a and b are not equal to 0. Let us check how we can use the substitution method to find the solution to a linear equation. In this method, we will substitute different values of the variable in the equation to check if we get both the left hand side and the right hand side as the same. Let us take an example.
Example
Solve the linear equation 3x + 6 = 15
Solution
We have been given the linear equation 3x + 6 = 15 and we need to find the value of x that satisfies this equation.
Let us start by substituting x = 1 in the L.H.S. we will have,
3x + 6 = 3 ( 1 ) + 6 = 3 + 6 = 9 . . . . . . . . . . . . . . ( 1 )
Now, the R.H.S of the equation is 15 and 9 ≠ 15. Hence, x = 1 is not the solution of the given equation.
Now, let us substitute x = 2 in the L.H.S. we will have,
3x + 6 = 3 ( 2 ) + 6 = 6 + 6 = 12 . . . . . . . . . . . . . . ( 2 )
Now, the R.H.S of the equation is 15 and 12 ≠ 15. Hence, x = 2 is not the solution of the given equation.
Next, let us substitute x = 3 in the L.H.S. we will have,
3x + 6 = 3 ( 3 ) + 6 = 9 + 6 = 15 . . . . . . . . . . . . . . ( 3 )
Now, the R.H.S of the equation is 15 and 15 = 15.
So, from ( 1 ), ( 2 ) and ( 3 ), we can say that x = 3 is the solution of the given equation.
Now, let us learn how we can use the substitution method to solve a system of equations.
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A system of linear equations is a system where we have more than one equation and variable and we need to find a solution that satisfies all the given equations. For example, consider the following system of equations –
2x + 5y =10
4x + 3y = 17
Above we have two linear equations in two variables, x and y. In order to find the solution to these equations, we must find a value each for x and y that satisfies both the equations. How can we use the substitution method to solve such a pair of linear equations? The followings steps are used to solve a system of linear equations through substitution –
Let us take an example.
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Example
Solve the following pair of linear equations
x – y = – 3
2x + 3y = 14
Solution
We have been the following system of linear equations –
x – y = – 3 . . . . . . . . . . . . . . . . . . . . ( 1 )
2x + 3y = 14 . . . . . . . . . . . . . . . . . . . ( 2 )
Let us use the substitution method to solve the above equations.
First, let us find the value of x from the first equation. We will have,
x – y = – 3
⇒ x = y – 3 . . . . . . . . . . . . . . . . . . . ( 3 )
Now, we will use this value of x and substitute it in equation ( 2 ). We will have,
2x + 3y = 14
⇒ 2 ( y – 3 ) + 3y = 14
⇒ 2y – 6 + 3y = 14
⇒ 5y – 6 = 14
⇒ 5y = 14 + 6
⇒ 5y = 20
⇒ y = 4 . . . . . . . . . . . . . . . . . . . ( 4 )
From ( 4 ), we now know that y = 4. Substituting this value of y in equation ( 3 ), we have,
x = y – 3
⇒ x = 4 – 3 = 1 . . . . . . . . . . . . ( 5 )
From ( 4 ) and ( 5 ), we can see that x = 1 and y = 4 is the solution to the given system of equations.
We can verify our result by substituting the values of x and y in equations ( 1 ) and ( 2 ). We will get
x – y = – 3
L.H.S
1 – ( 4 )
= 1 – 4
= – 3 = R. H.S . . . . . . . . . (6)
Also,
2x + 3y = 14
L.H.S
2x + 3y
= 2 ( 1 ) + 3 ( 4 )
= 2 + 12
= 14 = R.H.S . . . . . . . . . . ( 7 )
From ( 6 ) and ( 7 ) we can see that the value of x and y satisfy both the equation, hence our result is true.
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The substitution method is not the only method that is used to solve a system of equations. There is another method called the elimination method. In this method, we solve the two equations simultaneously and eliminate one of the variables. The major differences between these two methods are –
| Substitution Method | Elimination Method |
| In this method, we find the values of one variable in terms of the other variable. | In this method, we solve the two equations simultaneously and eliminate one of the variables. |
| Substitution method is more useful when we have simple equations, such as x = b + ay or y = ax + b | The elimination method can be conveniently used for any type of equation, especially when the coefficient of one of the variables in both equations is the same. |
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In algebra, the substitution method is the method that is used to solve a system of equations and find the values of the given variable that satisfy the equations. In this system, we use one equation to find the value of one variable in terms of another variable. This value is then substituted in the second equation. The equation thus obtained is a linear equation in one variable. This equation can be solved and a solution to one variable is thus obtained.
The substitution method is used when we have two linear equations in two variables.
In the substitution method, we find the value of one variable in terms of the other variable. This value is then used to substitute the value of the variable in the second equation so that we get a linear equation in one variable which can be solved easily.
In both methods, first, the value of one variable is obtained which is then used to find the value of the second variable.
The first step in the substitution method is that we use one equation to find the value of one variable in terms of another variable. For example, if we have an equation, x + 3y = 7, then we can write is as x = 7 – 3y.
The following are the steps involved in the substitution method –
a) First, we use one equation to find the value of one variable in terms of another variable.
b) This value is then substituted in the second equation
c) The equation thus obtained is a linear equation in one variable. This equation can be solved and a solution to one variable is thus obtained.
d) Using the solution obtained in the above step, the solution to the other variable is obtained by substitution the value in the first equation.
e) Finally, we have values of both the variables that satisfy the given system of equations.
In a linear system of equations, we use one equation to find the value of one variable in terms of another variable. This value is then substituted in the second equation. The equation thus obtained is a linear equation in one variable. This equation can be solved and a solution to one variable is thus obtained.
No, the substitution method is not restricted to linear equations only. It can be used for other equations as well such as quadratic equations or functions etc.