Substitution Method in Algebra Definition, Methods & Examples

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Updated: Dec 15, 2024

Lesson

Lesson Overview

What Is a Substitution Method In Math?
Solving Systems of Equations with Substitution
Key Differences Between Substitution and Elimination
Examples Of Substitution Method
Substitution Method Questions With Solutions

Systems of equations involve two or more equations with multiple variables. The Substitution Method in math is a key technique for solving these systems by replacing one variable with an equivalent expression derived from the other equation.

This simplification allows us to isolate and solve for each unknown, effectively finding the solution that satisfies all equations.

What Is a Substitution Method In Math?

The Substitution Method is a technique for solving systems of equations. It involves solving one equation for one variable, then substituting that expression into the other equation. This reduces the system to a single equation with one variable, making it solvable.

Example:

Solve the system:

x + y = 5

x - y = 1

Solve for x in the first equation: x = 5 - y

Substitute (5 - y) for x in the second equation: (5 - y) - y = 1

Simplify and solve for y: 5 - 2y = 1 => y = 2

Substitute the value of y back into either original equation to find x: x + 2 = 5 => x = 3

Therefore, the solution is x = 3, y = 2.

Solving Systems of Equations with Substitution

The substitution method provides a structured approach to solving systems of equations. By strategically substituting variables, we can effectively unravel the interconnected relationships and determine the solution.

Simplify if needed: Get rid of any parentheses.
Solve for one variable: Pick an equation and get one variable by itself.
Substitute: Plug the expression from step 2 into the other equation.
Solve the new equation: Find the value of one variable.
Find the other variable: Plug the answer from step 4 back into either of the original equations.

Example:

Solve the system:

2x - y = 3
x + 2y = 14

No parentheses to simplify.

Solve the second equation for x since it has a coefficient of 1: x = 14 - 2y

Substitute (14 - 2y) for x in the first equation: 2(14 - 2y) - y = 3

Simplify and solve for y: 28 - 4y - y = 3 28 - 5y = 3 -5y = -25 y = 5

Substitute y = 5 back into either original equation. Let's use x + 2y = 14: x + 2(5) = 14 x + 10 = 14 x = 4

Therefore, the solution to the system of equations is x = 4, y = 5.

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Solving Systems with Substitution

Key Differences Between Substitution and Elimination

While both methods achieve the same goal, understanding their differences allows you to choose the most efficient approach for a given system of equations.

Feature	Substitution Method	Elimination Method
Core Idea	Isolate one variable and substitute it into the other equation.	Manipulate equations to eliminate one variable when they are added or subtracted.
When it's best	One equation is easily solved for one variable.	Coefficients of one variable are opposites or easily made into opposites.
First Step	Solve one equation for one variable.	Multiply one or both equations by a constant if needed.
Intermediate Step	Substitute to get a single equation with one variable.	Add or subtract the equations to eliminate one variable.
Complexity	Can become complex with fractions or decimals.	Generally simpler with whole numbers.

Comparison of Substitution and Elimination Methods for Solving Systems of Equations

Aspect	Substitution Method	Elimination Method
Formula	ax + by = cdx + ey = f	ax + by = cdx + ey = f
Steps	1. Solve one equation for a variable.2. Substitute into the second equation.3. Solve for the remaining variable.4. Substitute back to find the other variable.	1. Multiply equations to align the coefficients.2. Add or subtract to eliminate one variable.3. Solve for the remaining variable.4. Substitute back to find the other variable.
Example	Given:2x + y = 10x − y = 2Step 1: Solve x − y = 2 for x:x = y + 2Step 2: Substitute x = y + 2 into 2 x + y = 10:2(y + 2) + y = 10Step 3: Simplify and solve:3y = 6,y = 2.Step 4: Substitute y = 2 into x = y + 2:x = 4.	Given:2x + 3y = 64x − 3y = 12Step 1: Add the equations to eliminate y:(2x + 3y) + (4x − 3y) = 6 + 12,6x = 18.Step 2: Solve for x:X = 3.Step 3: Substitute x = 3 into 2x + 3y = 6:6 + 3y = 6,3y = 0,y = 0.
Result	x = 4, y = 2	x = 3, y = 0
Best Used When	One variable can easily be solved for.	Both equations can be manipulated to eliminate one variable easily.
Advantages	- Easy when one variable is easily isolated.- Works well for simple equations.	- Faster when both equations are set up for elimination.- No need for substitution of expressions.

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What do you know about substitution & evaluating expressions?

Examples Of Substitution Method

Example 1:

x + y = 7
x - y = 1

Solve: x = 7 - y

Substitute: (7 - y) - y = 1

Solve: 7 - 2y = 1 => y = 3

Substitute & Solve: x + 3 = 7 => x = 4

Solution: x = 4, y = 3

Example 2:

2x + y = 9
x - y = 3

Solve: x = 3 + y

Substitute: 2(3 + y) + y = 9

Solve: 6 + 3y = 9 => y = 1

Substitute & Solve: x - 1 = 3 => x = 4

Solution: x = 4, y = 1

Example 3:

3x + 2y = 11
x - y = 2

Solve: x = 2 + y

Substitute: 3(2 + y) + 2y = 11

Solve: 6 + 5y = 11 => y = 1

Substitute & Solve: x - 1 = 2 => x = 3

Solution: x = 3, y = 1

Example 4:

x + 2y = 8
2x - 3y = -5

Solve: x = 8 - 2y

Substitute: 2(8 - 2y) - 3y = -5

Solve: 16 - 7y = -5 => y = 3

Substitute & Solve: x + 2(3) = 8 => x = 2

Solution: x = 2, y = 3

Example 5:

5x - y = 13
2x + 3y = 12

Solve: y = 5x - 13

Substitute: 2x + 3(5x - 13) = 12

Solve: 17x - 39 = 12 => x = 3

Substitute & Solve: 5(3) - y = 13 => y = 2

Solution: x = 3, y = 2

Substitution Method Questions With Solutions

1: Solve the system of equations:

y = 2x - 1
3x + y = 11

Solution:

Since y is already isolated in the first equation, substitute (2x - 1) for y in the second equation: 3x + (2x - 1) = 11
Solve: 5x - 1 = 11 => 5x = 12 => x = 12/5
Substitute & Solve: y = 2(12/5) - 1 => y = 19/5

Answer: x = 12/5, y = 19/5

2: Solve the system of equations: