y-(y-1/2) = (y-2/3)-y+4 ( y-1 is the numerator and 2 is the denominator. Same with y-2/3)

We are given the equation:

\[
y - \left( y - \frac{1}{2} \right) = \left( y - \frac{2}{3} \right) - y + 4
\]

Let's simplify both sides step by step.

### Left-hand side:
\[
y - \left( y - \frac{1}{2} \right)
\]
Distribute the negative sign:
\[
y - y + \frac{1}{2} = \frac{1}{2}
\]

### Right-hand side:
\[
\left( y - \frac{2}{3} \right) - y + 4
\]
Distribute the terms:
\[
y - \frac{2}{3} - y + 4
\]
The \(y\) and \(-y\) cancel out:
\[
-\frac{2}{3} + 4
\]
Express 4 as a fraction with denominator 3:
\[
-\frac{2}{3} + \frac{12}{3} = \frac{10}{3}
\]

### Final equation:
Now we have:
\[
\frac{1}{2} = \frac{10}{3}
\]

This equation is false because \(\frac{1}{2}\) is not equal to \(\frac{10}{3}\). Therefore, there is no solution to this equation; it has no solutions.

Would you like more details, or do you have any questions?

Here are 5 related questions:
1. How can we handle equations with fractions more easily?
2. How can you determine if an equation has no solution?
3. What does it mean when variables cancel out in an equation?
4. Can equations have more than one solution? How can we find them?
5. What are some strategies for solving complex algebraic equations?

**Tip**: When simplifying equations with fractions, it's often helpful to clear the fractions by multiplying through by the least common denominator (LCD).

y - (y - 1/2) = (y - 2/3) - y + 4 ( y-1 is the numerator and 2 is the denominator. Same with y-2/3)

Discover how to solve a linear equation with fractions, demonstrating why the equation y - (y - 1/2) = (y - 2/3) - y + 4 has no solution. Learn step-by-step simplifications, including fraction manipulation and recognizing inconsistent equations. Suitable for Grades 8-10.

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