Solving and Verifying the Equation 2y 5/3 26/3 - y

In this article, we will walk through the process of solving the algebraic equation (2y frac{5}{3} frac{26}{3} - y) and then verify the solution. This step-by-step guide provides a clear understanding of the algebraic manipulation involved in solving such equations.

Step 1: Isolate the Variable (y)

Let's start by moving all terms involving y to one side of the equation and all constant terms to the other side. We do this by adding y to both sides of the equation:

(2y frac{5}{3} y frac{26}{3} - y y)

This simplifies to:

(3y frac{5}{3} frac{26}{3})

Next, subtract (frac{5}{3}) from both sides:

(3y frac{26}{3} - frac{5}{3})

This simplifies to:

(3y frac{26 - 5}{3} frac{21}{3})

Further simplifying gives:

(3y 7)

Step 2: Solve for (y)

Now, divide both sides by 3:

(y frac{7}{3})

Step 3: Check the Solution

To ensure the solution (y frac{7}{3}) is correct, substitute it back into the original equation:

Substitute y into the left side: 2(left(frac{7}{3}right)) - (frac{5}{3}) (frac{14}{3}) - (frac{5}{3}) (frac{19}{3}) Substitute y into the right side: (frac{26}{3}) - (left(frac{7}{3}right)) (frac{26 - 7}{3}) (frac{19}{3})

Since both sides are equal:
(frac{19}{3} frac{19}{3})

The solution (y frac{7}{3}) is verified to be correct.

Alternative Method: Simplifying the Equation

Given that (2y frac{5}{3} frac{26}{3} - y), we can multiply both sides by 3 to eliminate the fractions:

(3(2y frac{5}{3}) 3(frac{26}{3} - y))

This simplifies to:

(6y 5 26 - 3y)

Adding (3y) to both sides:

(9y 5 26)

Subtracting 5 from both sides:

(9y 21)

Dividing by 9:

(y frac{21}{9} frac{7}{3})

Verification:

Left side: 2(left(frac{7}{3}right)) - (frac{5}{3}) (frac{14}{3}) - (frac{5}{3}) (frac{19}{3}) Right side: (frac{26}{3}) - (left(frac{7}{3}right)) (frac{26 - 7}{3}) (frac{19}{3})

So (y frac{7}{3}) satisfies the given equation.

Adding Same Number Keeps the Equation Satisfied

The principle of adding the same number to both sides of an equation is a fundamental concept in algebra. This principle holds true for addition, subtraction, multiplication, and division. It is a cornerstone in solving linear equations. Dividing by a non-zero divisor is another critical step in solving equations.

Conclusion

By following the steps outlined in this article, you have successfully solved and verified the equation (2y frac{5}{3} frac{26}{3} - y). The solution is (y frac{7}{3}).

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