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title: [Math Basics] Linear Equations with Two Variables
date: 2022-02-15 14:23:14
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category:

• Mathematics
  tags:
• Mathematics
• Basics
• Linear Equations with Two Variables

Linear Equations with Two Variables#

An equation with two unknowns (variables) and the highest power of the unknown terms is 1 (linear) is called a linear equation with two variables. A combination of multiple equations is called a system of equations.

They generally look like this:

Equation 1: a₁x + b₁y = c₁

Equation 2: a₂x + b₂y = c₂

Examples:

Equation 1: 3x + 4y = 7

Equation 2: 4x + 6y = 9

Where a₁, a₂, b₁, b₂, c₁, c₂ are all constants.

Solving#

Determining if there is a solution#

• a₁/a₂ ≠ b₁/b₂, there is a unique solution
• a₁/a₂ = b₁/b₂ ≠ c₁/c₂, no solution
• a₁/a₂ = b₁/b₂ = c₁/c₂, infinitely many solutions

Solving the equations#

1. Substitution method
2. Addition/subtraction method

Substitution method#

By substituting one unknown with a value, we can simplify one of the equations to have only one unknown.

Find the real solutions for the following system of equations:

Equation 1: x + y = 3

Equation 2: x / y = 5

1. Equation 1 simplifies to x = 3 - y
2. Substitute the simplified Equation 1 into Equation 2: (3 - y) / y = 5
3. 3 - y = 5y
4. y = 1 / 2
5. When y = 1 / 2, substitute the value of y into the original Equation 1: x + (1 / 2) = 3
6. x + (1 / 2) = 6 / 2
7. x = 5 / 2

Addition/subtraction method#

Treat the system of equations as a whole, regardless of the number of equations, it can be understood as A = B.

Equation 1: 2x + y = 7, can be understood as A = B

Equation 2: 4x - 2y = 6, similarly can be understood as A = B

By performing addition, subtraction, multiplication, and division operations on the system as a whole, eliminate one unknown to have only one unknown in the equation.

1. Multiply Equation 1 by 2: 4x + 2y = 14
2. Subtract Equation 1 from Equation 2: (4x - 2y) - (4x + 2y) = 6 - 14
3. 4x - 2y - 4x - 2y = -8
4. -4y = -8
5. y = 2
6. Substitute y = 2 into Equation 2: 4x - 2 * 2 = 6, which simplifies to 4x - 10 = 0
7. x = 5 / 2

Tips#

When both equations have integer constant terms, it is recommended to use the addition/subtraction method. When the constant terms are fractions, the substitution method is recommended.

Also, it is advisable to eliminate the unknown that is not fixed first, and it is recommended to substitute the values back into the original equations after solving, as there may be errors along the way that can be discovered and resolved during the back substitution process.

There are a total of 35 chickens and rabbits, with a total of 94 legs. How many chickens and rabbits are there?

Solution: From the given information, we have the following system of equations:

Equation 1: x + y = 35

Equation 2: 2x + 4y = 94

1. Multiply Equation 1 by 2: 2x + 2y = 70
2. Subtract Equation 1 from Equation 2: (2x + 4y) - (2x + 2y) = 94 - 70
3. 2x + 4y - 2x - 2y = 24
4. 2y = 24
5. y = 12
6. Since y = 12, substitute it into Equation 1: x + 12 = 35
7. x = 35 - 12, x = 23

Equation 1: x + 2y = 4

Equation 2: 2x + ay = 2a

If the system of equations has infinitely many solutions, what is the value of a? When a = 6, find the solution to the system of equations.

Solution:

Using the determinant a₁/a₂ = b₁/b₂ = c₁/c₂, we have: 1/2 = 2/a = 4/2a, solving for a = 4.

When a = 6, the original system of equations becomes:

Equation 1: x + 2y = 4

Equation 2: 2x + 6y = 12

1. Using the addition/subtraction method, multiply Equation 1 by 2: 2x + 4y = 8
2. Subtract Equation 1 from Equation 2: 2x + 6y - (2x + 4y) = 12 - 8
3. 2x + 6y - 2x - 4y = 4
4. 2y = 4
5. y = 2
6. Substitute y = 2 into Equation 1: x + 2 * 2 = 4, which simplifies to x = 0.