What types of solutions are available for the equations 3x + 2y = 4 and 2x + y = 5?

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The equations you provided are:

  • 3x + 2y = 4
  • 2x + y = 5

To determine the solutions, we can solve this system of linear equations using either the substitution method or the elimination method. Let’s use the substitution method for clarity.

Step 1: Solve one equation for one variable

We can solve the second equation for y:

2x + y = 5
=> y = 5 - 2x

Step 2: Substitute into the first equation

Now substitute this expression for y into the first equation:

3x + 2(5 - 2x) = 4
3x + 10 - 4x = 4

Step 3: Solve for x

Combining like terms gives us:

-x + 10 = 4
-x = 4 - 10
-x = -6
x = 6

Step 4: Find y

Now, substitute x = 6 back into the equation we derived for y:

y = 5 - 2(6)
y = 5 - 12
y = -7

Conclusion

The solution to the system of equations is:

(x, y) = (6, -7)

This means that the only point where both lines intersect is at (6, -7). Therefore, the system of equations has a single solution, indicating that the two lines represented by these equations are not parallel and meet at exactly one point.

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