Math Problem Statement

Find the intersection of the two lines: 2x + 3y = 5 and 2x + 3y = 10.

Solution

The image contains a system of linear equations:

  1. 2x+3y=52x + 3y = 5
  2. 2x+3y=102x + 3y = 10

Let's solve this system to find the point of intersection, if it exists.

Step 1: Analyze the system

Both equations appear to have the same left-hand side but different right-hand sides:

2x+3y=5(Equation 1)2x + 3y = 5 \quad \text{(Equation 1)} 2x+3y=10(Equation 2)2x + 3y = 10 \quad \text{(Equation 2)}

Step 2: Subtract one equation from the other

Subtract Equation 1 from Equation 2 to eliminate xx and yy:

(2x+3y)(2x+3y)=105(2x + 3y) - (2x + 3y) = 10 - 5 0=50 = 5

Step 3: Interpretation

The result 0=50 = 5 is a contradiction, which indicates that there is no solution. This system of equations is inconsistent, meaning the lines are parallel and do not intersect.

Conclusion

The system of equations does not have a point of intersection. The two lines represented by these equations are parallel.

Would you like more details on this solution or have any other questions?

Here are 5 related questions you might find interesting:

  1. How can we tell if two lines are parallel from their equations?
  2. What are the possible outcomes when solving a system of linear equations?
  3. How do we find the point of intersection for consistent systems of equations?
  4. What methods can be used to solve a system of equations besides substitution and elimination?
  5. How can we graphically represent parallel lines?

Tip: When two equations in a system have the same coefficients for all variables but different constants, the system is inconsistent, leading to parallel lines with no intersection.

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Math Problem Analysis

Mathematical Concepts

Linear Equations
Systems of Equations
Parallel Lines

Formulas

Standard form of a linear equation: Ax + By = C

Theorems

If two linear equations have the same coefficients for variables but different constants, the lines are parallel.

Suitable Grade Level

Grades 8-10