Solved on Sep 19, 2023

Solve the equation $\frac{4}{y-1}-\frac{5}{y+2}=\frac{3}{y}$ and give the answers as integers or simplified fractions.

STEP 1

Assumptions1. The variable y is not equal to0,1, or -, as these values would make the denominators of the fractions equal to zero, which is undefined in mathematics.

STEP 2

The first step in solving this equation is to find a common denominator for all the fractions. The common denominator is the product of all the different denominators, which in this case is $(y-1)(y+2)y$ .

STEP 3

Multiply each term by the common denominator.
$(y+2)y \cdot \frac{}{y-1} - (y-1)y \cdot \frac{5}{y+2} = (y-1)(y+2) \cdot \frac{3}{y}$

STEP 4

implify each term by cancelling out the common factors.
$4y(y+2) -y(y-1) =3(y-1)(y+2)$

STEP 5

Expand each term.
$4y^2 +8y -5y^2 +5y =3y^2 -3$

STEP 6

Combine like terms on each side of the equation.
$-y^2 +13y =3y^2 -3$

STEP 7

Rearrange the equation to set it equal to zero.
$4y^2 -13y +3 =0$

STEP 8

This is a quadratic equation in the form of $ay^2 + by + c =0$ . We can solve it using the quadratic formula, which is $y = \frac{-b \pm \sqrt{b^2 -4ac}}{2a}$ .

STEP 9

Substitute the coefficients a, b, and c from the equation into the quadratic formula.
$y = \frac{-(-13) \pm \sqrt{(-13)^2 -443}}{2*4}$

STEP 10

implify the expression under the square root.
$y = \frac{13 \pm \sqrt{169 -48}}{8}$

STEP 11

implify the expression further.
$y = \frac{13 \pm \sqrt{121}}{8}$

STEP 12

Take the square root of121.
$y = \frac{ \pm11}{8}$

STEP 13

olve for y by using the plus and minus values separately.
$y = \frac{13 +11}{8} \quad or \quad y = \frac{13 -11}{8}$

STEP 14

implify the fractions to get the solutions.
$y = \frac{24}{8} \quad or \quad y = \frac{2}{8}$

STEP 15

implify the fractions to their lowest terms to get the final solutions.
$y =3 \quad or \quad y = \frac{}{4}$ The solutions to the equation are $y =3$ and $y = \frac{}{4}$ .

Was this helpful?

Solve the equation $\frac{4}{y-1}-\frac{5}{y+2}=\frac{3}{y}$ and give the answers as integers or simplified fractions.

STEP 1

Assumptions1. The variable y is not equal to0,1, or -, as these values would make the denominators of the fractions equal to zero, which is undefined in mathematics.

STEP 2

The first step in solving this equation is to find a common denominator for all the fractions. The common denominator is the product of all the different denominators, which in this case is $(y-1)(y+2)y$ .

STEP 3

Multiply each term by the common denominator.
$(y+2)y \cdot \frac{}{y-1} - (y-1)y \cdot \frac{5}{y+2} = (y-1)(y+2) \cdot \frac{3}{y}$

STEP 4

implify each term by cancelling out the common factors.
$4y(y+2) -y(y-1) =3(y-1)(y+2)$

STEP 5

Expand each term.
$4y^2 +8y -5y^2 +5y =3y^2 -3$

STEP 6

Combine like terms on each side of the equation.
$-y^2 +13y =3y^2 -3$

STEP 7

Rearrange the equation to set it equal to zero.
$4y^2 -13y +3 =0$

STEP 8

This is a quadratic equation in the form of $ay^2 + by + c =0$ . We can solve it using the quadratic formula, which is $y = \frac{-b \pm \sqrt{b^2 -4ac}}{2a}$ .

STEP 9

Substitute the coefficients a, b, and c from the equation into the quadratic formula.
$y = \frac{-(-13) \pm \sqrt{(-13)^2 -443}}{2*4}$

STEP 10

implify the expression under the square root.
$y = \frac{13 \pm \sqrt{169 -48}}{8}$

STEP 11

implify the expression further.
$y = \frac{13 \pm \sqrt{121}}{8}$

STEP 12

Take the square root of121.
$y = \frac{ \pm11}{8}$

STEP 13

olve for y by using the plus and minus values separately.
$y = \frac{13 +11}{8} \quad or \quad y = \frac{13 -11}{8}$

STEP 14

implify the fractions to get the solutions.
$y = \frac{24}{8} \quad or \quad y = \frac{2}{8}$

STEP 15

implify the fractions to their lowest terms to get the final solutions.
$y =3 \quad or \quad y = \frac{}{4}$ The solutions to the equation are $y =3$ and $y = \frac{}{4}$ .

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Solve the equation 4y−1−5y+2=3y\frac{4}{y-1}-\frac{5}{y+2}=\frac{3}{y}y−14​−y+25​=y3​ and give the answers as integers or simplified fractions.

STEP 1

Assumptions1. The variable y is not equal to0,1, or -, as these values would make the denominators of the fractions equal to zero, which is undefined in mathematics.

STEP 2

The first step in solving this equation is to find a common denominator for all the fractions. The common denominator is the product of all the different denominators, which in this case is (y−1)(y+2)y(y-1)(y+2)y(y−1)(y+2)y.

STEP 3

Multiply each term by the common denominator.(y+2)y⋅y−1−(y−1)y⋅5y+2=(y−1)(y+2)⋅3y(y+2)y \cdot \frac{}{y-1} - (y-1)y \cdot \frac{5}{y+2} = (y-1)(y+2) \cdot \frac{3}{y} (y+2)y⋅y−1​−(y−1)y⋅y+25​=(y−1)(y+2)⋅y3​

STEP 4

implify each term by cancelling out the common factors.4y(y+2)−y(y−1)=3(y−1)(y+2)4y(y+2) -y(y-1) =3(y-1)(y+2) 4y(y+2)−y(y−1)=3(y−1)(y+2)

STEP 5

Expand each term.4y2+8y−5y2+5y=3y2−34y^2 +8y -5y^2 +5y =3y^2 -34y2+8y−5y2+5y=3y2−3

STEP 6

Combine like terms on each side of the equation.−y2+13y=3y2−3-y^2 +13y =3y^2 -3−y2+13y=3y2−3

STEP 7

Rearrange the equation to set it equal to zero.4y2−13y+3=04y^2 -13y +3 =04y2−13y+3=0

STEP 8

This is a quadratic equation in the form of ay2+by+c=0ay^2 + by + c =0ay2+by+c=0. We can solve it using the quadratic formula, which is y=−b±b2−4ac2ay = \frac{-b \pm \sqrt{b^2 -4ac}}{2a}y=2a−b±b2−4ac​​.

STEP 9

Substitute the coefficients a, b, and c from the equation into the quadratic formula.y=−(−13)±(−13)2−4∗4∗32∗4y = \frac{-(-13) \pm \sqrt{(-13)^2 -4*4*3}}{2*4} y=2∗4−(−13)±(−13)2−4∗4∗3​​

STEP 10

implify the expression under the square root.y=13±169−488y = \frac{13 \pm \sqrt{169 -48}}{8} y=813±169−48​​

STEP 11

implify the expression further.y=13±1218y = \frac{13 \pm \sqrt{121}}{8} y=813±121​​

STEP 12

Take the square root of121.y=±118y = \frac{ \pm11}{8} y=8±11​

STEP 13

olve for y by using the plus and minus values separately.y=13+118ory=13−118y = \frac{13 +11}{8} \quad or \quad y = \frac{13 -11}{8} y=813+11​ory=813−11​

STEP 14

implify the fractions to get the solutions.y=248ory=28y = \frac{24}{8} \quad or \quad y = \frac{2}{8} y=824​ory=82​

STEP 15

implify the fractions to their lowest terms to get the final solutions.y=3ory=4y =3 \quad or \quad y = \frac{}{4} y=3ory=4​The solutions to the equation are y=3y =3y=3 and y=4y = \frac{}{4}y=4​.

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Solve the equation $\frac{4}{y-1}-\frac{5}{y+2}=\frac{3}{y}$ and give the answers as integers or simplified fractions.

The first step in solving this equation is to find a common denominator for all the fractions. The common denominator is the product of all the different denominators, which in this case is $(y-1)(y+2)y$ .

Multiply each term by the common denominator.
$(y+2)y \cdot \frac{}{y-1} - (y-1)y \cdot \frac{5}{y+2} = (y-1)(y+2) \cdot \frac{3}{y}$

implify each term by cancelling out the common factors.
$4y(y+2) -y(y-1) =3(y-1)(y+2)$

Expand each term.
$4y^2 +8y -5y^2 +5y =3y^2 -3$

Combine like terms on each side of the equation.
$-y^2 +13y =3y^2 -3$

Rearrange the equation to set it equal to zero.
$4y^2 -13y +3 =0$

This is a quadratic equation in the form of $ay^2 + by + c =0$ . We can solve it using the quadratic formula, which is $y = \frac{-b \pm \sqrt{b^2 -4ac}}{2a}$ .

Substitute the coefficients a, b, and c from the equation into the quadratic formula.
$y = \frac{-(-13) \pm \sqrt{(-13)^2 -443}}{2*4}$

implify the expression under the square root.
$y = \frac{13 \pm \sqrt{169 -48}}{8}$

implify the expression further.
$y = \frac{13 \pm \sqrt{121}}{8}$

Take the square root of121.
$y = \frac{ \pm11}{8}$

olve for y by using the plus and minus values separately.
$y = \frac{13 +11}{8} \quad or \quad y = \frac{13 -11}{8}$

implify the fractions to get the solutions.
$y = \frac{24}{8} \quad or \quad y = \frac{2}{8}$

implify the fractions to their lowest terms to get the final solutions.
$y =3 \quad or \quad y = \frac{}{4}$ The solutions to the equation are $y =3$ and $y = \frac{}{4}$ .