How do you find a pair of linear equations?

Finding a pair of linear equations usually means you're trying to either create them from given information or determine if a set of equations is a pair of linear equations. This answer addresses both scenarios.

Scenario 1: Creating a Pair of Linear Equations

This typically involves being given two pieces of information about a relationship between two variables (often x and y). Here's a breakdown:

1. Identify the variables: Determine the two quantities you want to relate with your equations. Call them x and y.

2. Translate the information into equations: Look for keywords or phrases that translate to mathematical operations. Common examples include:

   • "The sum of..." implies addition.
   • "The difference between..." implies subtraction.
   • "Is equal to..." implies equality (=).
   • "Twice the value of..." implies multiplication by 2.

3. Formulate the equations: Write two separate linear equations using the information you've gathered. A linear equation has the general form: ax + by = c, where a, b, and c are constants.

Example:

Suppose you're told:

• "The sum of two numbers is 10."
• "The difference between the same two numbers is 4."

Let x be the first number and y be the second number. Then the pair of linear equations is:

• x + y = 10
• x - y = 4

Scenario 2: Determining if a Set of Equations is a Pair of Linear Equations

A "pair of linear equations" simply means you have two equations where the variables are raised to the power of 1 (no exponents, square roots, etc.). To confirm, check these points:

1. Two Equations: Make sure you have exactly two equations.

2. Linearity: Ensure each equation is linear. This means:

   • The variables (usually x and y) have no exponents other than 1.
   • The variables are not inside any other functions (like square roots, trigonometric functions, logarithms, etc.).
   • The variables are not multiplied together (no xy terms).

Example:

• 2x + 3y = 7 and x - y = 1 are a pair of linear equations.

• x^2 + y = 5 and x + y = 2 are not a pair of linear equations because the first equation has x^2 making it a quadratic equation.

Consistency and Dependency

Once you have a pair of linear equations, you can analyze their properties:

• Consistent: The equations have at least one solution (they intersect at one or more points).
• Inconsistent: The equations have no solution (they are parallel lines).
• Dependent and Consistent: The equations represent the same line (they have infinitely many solutions).

These relationships can be determined by comparing the coefficients of the equations:

Let the two linear equations be:

a1x + b1y = c1
a2x + b2y = c2

Condition	Result
a1/a2 ≠ b1/b2	Consistent (unique solution)
a1/a2 = b1/b2 ≠ c1/c2	Inconsistent (no solution)
a1/a2 = b1/b2 = c1/c2	Dependent and Consistent (infinitely many solutions)

In summary, finding a pair of linear equations involves either constructing them from given information by translating relationships into mathematical expressions or verifying that a given set of equations meets the criteria for linearity (variables to the power of 1, no unusual functions).