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Math  /  Algebra

Questionyy−1+1y−4=y2y2−5y+4\frac{y}{y-1}+\frac{1}{y-4}=\frac{y^{2}}{y^{2}-5 y+4}

Studdy Solution

STEP 1

1. The given equation involves rational expressions with polynomials.
2. The equation can be simplified by finding a common denominator.
3. The equation can be solved by equating the numerators after ensuring the denominators are the same.
4. Simplification might involve factoring polynomials and canceling common factors.

STEP 2

1. Simplify the right-hand side of the equation.
2. Find a common denominator for the left-hand side.
3. Combine the fractions on the left-hand side.
4. Set the numerators equal to each other.
5. Solve the resulting polynomial equation.

STEP 3

Factor the denominator on the right-hand side.
y2−5y+4=(y−1)(y−4) y^2 - 5y + 4 = (y-1)(y-4)

STEP 4

Express the right-hand side with the factored denominator.
y2(y−1)(y−4) \frac{y^2}{(y-1)(y-4)}

STEP 5

Find a common denominator for the left-hand side fractions yy−1\frac{y}{y-1} and 1y−4\frac{1}{y-4}.
Common Denominator=(y−1)(y−4) \text{Common Denominator} = (y-1)(y-4)

STEP 6

Rewrite the left-hand side fractions with the common denominator.
y(y−4)(y−1)(y−4)+1(y−1)(y−1)(y−4) \frac{y(y-4)}{(y-1)(y-4)} + \frac{1(y-1)}{(y-1)(y-4)}

STEP 7

Simplify the numerators of the fractions on the left-hand side.
y2−4y(y−1)(y−4)+y−1(y−1)(y−4) \frac{y^2 - 4y}{(y-1)(y-4)} + \frac{y-1}{(y-1)(y-4)}

STEP 8

Combine the fractions on the left-hand side.
(y2−4y)+(y−1)(y−1)(y−4) \frac{(y^2 - 4y) + (y - 1)}{(y-1)(y-4)}

STEP 9

Simplify the numerator of the combined fraction.
y2−4y+y−1=y2−3y−1 y^2 - 4y + y - 1 = y^2 - 3y - 1

STEP 10

Write the simplified left-hand side fraction.
y2−3y−1(y−1)(y−4) \frac{y^2 - 3y - 1}{(y-1)(y-4)}

STEP 11

Set the numerators equal to each other since the denominators are the same.
y2−3y−1=y2 y^2 - 3y - 1 = y^2

STEP 12

Subtract y2y^2 from both sides of the equation to simplify.
−3y−1=0 -3y - 1 = 0

STEP 13

Solve for yy.
−3y=1 -3y = 1 y=−13 y = -\frac{1}{3}
Solution: y=−13y = -\frac{1}{3}

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