Math Problem Statement

6x - 5y = 15 and -6x + 5y = -15

Solution

The given system of equations is:

6x5y=156x - 5y = 15 6x+5y=15-6x + 5y = -15

To solve this system, you can either use substitution or elimination. I will demonstrate the elimination method since the coefficients of xx and yy are already opposites.

Step 1: Add the two equations

(6x5y)+(6x+5y)=15+(15)(6x - 5y) + (-6x + 5y) = 15 + (-15)

The terms 6x6x and 6x-6x cancel out, as do 5y-5y and 5y5y. Therefore, we get:

0=00 = 0

Step 2: Interpret the result

The equation 0=00 = 0 is true for all values of xx and yy, meaning the system has infinitely many solutions. The two equations represent the same line, and any point on that line is a solution.

So, this system has infinitely many solutions.

Would you like further clarification on this? Here are some related questions to explore:

  1. How can you graph the solution to this system?
  2. How do you solve systems of equations using substitution?
  3. How do you interpret systems with no solutions?
  4. What does it mean geometrically when two equations represent the same line?
  5. What is the difference between dependent and independent systems?

Tip: Always check the result of adding or subtracting the equations in elimination method to see if you get a true or false statement. If it's true, the system has infinite solutions; if false, it has no solutions.

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Math Problem Analysis

Mathematical Concepts

Systems of Linear Equations
Elimination Method

Formulas

Elimination method: Adding or subtracting equations to eliminate variables

Theorems

Systems with infinitely many solutions (dependent systems)
Zero equals zero (indicating infinite solutions)

Suitable Grade Level

Grades 7-10