Solve y-(7-8y)/9y-(3+4y)=2/3 | Equatix - Math Solver

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Example

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3(r+2s)=2t-4

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Question

$$y-(7-8y)/9y-(3+4y)=2/3$$

Solve for y

$y = \frac{\sqrt{553} + 17}{8} \approx 5.064494004$
$y=\frac{17-\sqrt{553}}{8}\approx -0.814494004$

Steps Using the Quadratic Formula

Steps for Completing the Square

Steps Using the Quadratic Formula

Multiply both sides of the equation by $9$, the least common multiple of $9,3$.

$$9\left(y-\frac{7-8y}{9}y\right)-9\left(3+4y\right)=6$$

Express $\frac{7-8y}{9}y$ as a single fraction.

$$9\left(y-\frac{\left(7-8y\right)y}{9}\right)-9\left(3+4y\right)=6$$

Use the distributive property to multiply $7-8y$ by $y$.

$$9\left(y-\frac{7y-8y^{2}}{9}\right)-9\left(3+4y\right)=6$$

Use the distributive property to multiply $9$ by $y-\frac{7y-8y^{2}}{9}$.

$$9y+9\left(-\frac{7y-8y^{2}}{9}\right)-9\left(3+4y\right)=6$$

Express $9\left(-\frac{7y-8y^{2}}{9}\right)$ as a single fraction.

$$9y+\frac{-9\left(7y-8y^{2}\right)}{9}-9\left(3+4y\right)=6$$

Cancel out $9$ and $9$.

$$9y-\left(7y-8y^{2}\right)-9\left(3+4y\right)=6$$

To find the opposite of $7y-8y^{2}$, find the opposite of each term.

$$9y-7y-\left(-8y^{2}\right)-9\left(3+4y\right)=6$$

The opposite of $-8y^{2}$ is $8y^{2}$.

$$9y-7y+8y^{2}-9\left(3+4y\right)=6$$

Combine $9y$ and $-7y$ to get $2y$.

$$2y+8y^{2}-9\left(3+4y\right)=6$$

Use the distributive property to multiply $-9$ by $3+4y$.

$$2y+8y^{2}-27-36y=6$$

Combine $2y$ and $-36y$ to get $-34y$.

$$-34y+8y^{2}-27=6$$

Subtract $6$ from both sides.

$$-34y+8y^{2}-27-6=0$$

Subtract $6$ from $-27$ to get $-33$.

$$-34y+8y^{2}-33=0$$

All equations of the form $ax^{2}+bx+c=0$ can be solved using the quadratic formula: $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$. The quadratic formula gives two solutions, one when $±$ is addition and one when it is subtraction.

$$8y^{2}-34y-33=0$$

This equation is in standard form: $ax^{2}+bx+c=0$. Substitute $8$ for $a$, $-34$ for $b$, and $-33$ for $c$ in the quadratic formula, $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$.

$$y=\frac{-\left(-34\right)±\sqrt{\left(-34\right)^{2}-4\times 8\left(-33\right)}}{2\times 8}$$

Square $-34$.

$$y=\frac{-\left(-34\right)±\sqrt{1156-4\times 8\left(-33\right)}}{2\times 8}$$

Multiply $-4$ times $8$.

$$y=\frac{-\left(-34\right)±\sqrt{1156-32\left(-33\right)}}{2\times 8}$$

Multiply $-32$ times $-33$.

$$y=\frac{-\left(-34\right)±\sqrt{1156+1056}}{2\times 8}$$

Add $1156$ to $1056$.

$$y=\frac{-\left(-34\right)±\sqrt{2212}}{2\times 8}$$

Take the square root of $2212$.

$$y=\frac{-\left(-34\right)±2\sqrt{553}}{2\times 8}$$

The opposite of $-34$ is $34$.

$$y=\frac{34±2\sqrt{553}}{2\times 8}$$

Multiply $2$ times $8$.

$$y=\frac{34±2\sqrt{553}}{16}$$

Now solve the equation $y=\frac{34±2\sqrt{553}}{16}$ when $±$ is plus. Add $34$ to $2\sqrt{553}$.

$$y=\frac{2\sqrt{553}+34}{16}$$

Divide $34+2\sqrt{553}$ by $16$.

$$y=\frac{\sqrt{553}+17}{8}$$

Now solve the equation $y=\frac{34±2\sqrt{553}}{16}$ when $±$ is minus. Subtract $2\sqrt{553}$ from $34$.

$$y=\frac{34-2\sqrt{553}}{16}$$

Divide $34-2\sqrt{553}$ by $16$.

$$y=\frac{17-\sqrt{553}}{8}$$

The equation is now solved.

$$y=\frac{\sqrt{553}+17}{8}$$ $$y=\frac{17-\sqrt{553}}{8}$$

Steps for Completing the Square

Multiply both sides of the equation by $9$, the least common multiple of $9,3$.

$$9\left(y-\frac{7-8y}{9}y\right)-9\left(3+4y\right)=6$$

Express $\frac{7-8y}{9}y$ as a single fraction.

$$9\left(y-\frac{\left(7-8y\right)y}{9}\right)-9\left(3+4y\right)=6$$

Use the distributive property to multiply $7-8y$ by $y$.

$$9\left(y-\frac{7y-8y^{2}}{9}\right)-9\left(3+4y\right)=6$$

Use the distributive property to multiply $9$ by $y-\frac{7y-8y^{2}}{9}$.

$$9y+9\left(-\frac{7y-8y^{2}}{9}\right)-9\left(3+4y\right)=6$$

Express $9\left(-\frac{7y-8y^{2}}{9}\right)$ as a single fraction.

$$9y+\frac{-9\left(7y-8y^{2}\right)}{9}-9\left(3+4y\right)=6$$

Cancel out $9$ and $9$.

$$9y-\left(7y-8y^{2}\right)-9\left(3+4y\right)=6$$

To find the opposite of $7y-8y^{2}$, find the opposite of each term.

$$9y-7y-\left(-8y^{2}\right)-9\left(3+4y\right)=6$$

The opposite of $-8y^{2}$ is $8y^{2}$.

$$9y-7y+8y^{2}-9\left(3+4y\right)=6$$

Combine $9y$ and $-7y$ to get $2y$.

$$2y+8y^{2}-9\left(3+4y\right)=6$$

Use the distributive property to multiply $-9$ by $3+4y$.

$$2y+8y^{2}-27-36y=6$$

Combine $2y$ and $-36y$ to get $-34y$.

$$-34y+8y^{2}-27=6$$

Add $27$ to both sides.

$$-34y+8y^{2}=6+27$$

Add $6$ and $27$ to get $33$.

$$-34y+8y^{2}=33$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form $x^{2}+bx=c$.

$$8y^{2}-34y=33$$

Divide both sides by $8$.

$$\frac{8y^{2}-34y}{8}=\frac{33}{8}$$

Dividing by $8$ undoes the multiplication by $8$.

$$y^{2}+\left(-\frac{34}{8}\right)y=\frac{33}{8}$$

Reduce the fraction $\frac{-34}{8}$ to lowest terms by extracting and canceling out $2$.

$$y^{2}-\frac{17}{4}y=\frac{33}{8}$$

Divide $-\frac{17}{4}$, the coefficient of the $x$ term, by $2$ to get $-\frac{17}{8}$. Then add the square of $-\frac{17}{8}$ to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}-\frac{17}{4}y+\left(-\frac{17}{8}\right)^{2}=\frac{33}{8}+\left(-\frac{17}{8}\right)^{2}$$

Square $-\frac{17}{8}$ by squaring both the numerator and the denominator of the fraction.

$$y^{2}-\frac{17}{4}y+\frac{289}{64}=\frac{33}{8}+\frac{289}{64}$$

Add $\frac{33}{8}$ to $\frac{289}{64}$ by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y^{2}-\frac{17}{4}y+\frac{289}{64}=\frac{553}{64}$$

Factor $y^{2}-\frac{17}{4}y+\frac{289}{64}$. In general, when $x^{2}+bx+c$ is a perfect square, it can always be factored as $\left(x+\frac{b}{2}\right)^{2}$.

$$\left(y-\frac{17}{8}\right)^{2}=\frac{553}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-\frac{17}{8}\right)^{2}}=\sqrt{\frac{553}{64}}$$

Simplify.

$$y-\frac{17}{8}=\frac{\sqrt{553}}{8}$$ $$y-\frac{17}{8}=-\frac{\sqrt{553}}{8}$$

Add $\frac{17}{8}$ to both sides of the equation.

$$y=\frac{\sqrt{553}+17}{8}$$ $$y=\frac{17-\sqrt{553}}{8}$$