Determine whether the system of linear equations has one and only one solution, infinitely many solutions, or no solution.

5*x*

−

6*y*

=

6

10*x*

−

12*y*

=

12

To determine whether the system of linear equations has one solution, infinitely many solutions, or no solution, we can use the following approach:

1. First, write the given system of linear equations clearly:

\[
5x - 6y = 6 \tag{1}
\]
\[
10x - 12y = 12 \tag{2}
\]

2. Now, we will simplify and compare the equations. Notice that the second equation is a multiple of the first equation. Specifically:

\[
10x - 12y = 2 \cdot (5x - 6y)
\]

Thus, equation (2) is essentially twice equation (1). Dividing equation (2) by 2 gives us:

\[
5x - 6y = 6
\]

This is the same as equation (1), meaning the two equations are **dependent** and represent the same line.

### Conclusion:
Since the two equations represent the same line, the system has **infinitely many solutions**.

---

Would you like further details or have any questions?

Here are 5 related questions you might want to explore:

1. How can you identify parallel lines in a system of linear equations?
2. What are the conditions for a system of equations to have no solution?
3. How do you solve a system of linear equations using the substitution method?
4. What is the geometric interpretation of infinitely many solutions?
5. How does the determinant of a coefficient matrix affect the number of solutions?

**Tip:** For any system of two linear equations, if the equations are proportional, the system has infinitely many solutions or no solution, depending on whether the constants are also proportional.

Determine whether the system of linear equations has one and only one solution, infinitely many solutions, or no solution.

5x - 6y = 6
10x - 12y = 12

Learn how to determine if a system of linear equations, such as 5x - 6y = 6 and 10x - 12y = 12, has one solution, no solution, or infinitely many solutions. Understand key algebraic concepts like proportional equations and dependent systems. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation