Solve (3z + 5)/(9z + 2) = (3z + 4)/(4z + 7) | Equatix - Math Solver

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Example

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

a⋅(b-2)=3b

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

a⋅(b-2)=3b

Question

$$\frac{3z+5}{9z+2}=\frac{3z+4}{4z+7}$$

Solve for z

$z = \frac{\sqrt{1621} - 1}{30} \approx 1.308721476$
$z=\frac{-\sqrt{1621}-1}{30}\approx -1.375388143$

Steps Using the Quadratic Formula

Steps for Completing the Square

Steps Using the Quadratic Formula

Variable $z$ cannot be equal to any of the values $-\frac{7}{4},-\frac{2}{9}$ since division by zero is not defined. Multiply both sides of the equation by $\left(4z+7\right)\left(9z+2\right)$, the least common multiple of $9z+2,4z+7$.

$$\left(4z+7\right)\left(3z+5\right)=\left(9z+2\right)\left(3z+4\right)$$

Use the distributive property to multiply $4z+7$ by $3z+5$ and combine like terms.

$$12z^{2}+41z+35=\left(9z+2\right)\left(3z+4\right)$$

Use the distributive property to multiply $9z+2$ by $3z+4$ and combine like terms.

$$12z^{2}+41z+35=27z^{2}+42z+8$$

Subtract $27z^{2}$ from both sides.

$$12z^{2}+41z+35-27z^{2}=42z+8$$

Combine $12z^{2}$ and $-27z^{2}$ to get $-15z^{2}$.

$$-15z^{2}+41z+35=42z+8$$

Subtract $42z$ from both sides.

$$-15z^{2}+41z+35-42z=8$$

Combine $41z$ and $-42z$ to get $-z$.

$$-15z^{2}-z+35=8$$

Subtract $8$ from both sides.

$$-15z^{2}-z+35-8=0$$

Subtract $8$ from $35$ to get $27$.

$$-15z^{2}-z+27=0$$

This equation is in standard form: $ax^{2}+bx+c=0$. Substitute $-15$ for $a$, $-1$ for $b$, and $27$ for $c$ in the quadratic formula, $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$.

$$z=\frac{-\left(-1\right)±\sqrt{1-4\left(-15\right)\times 27}}{2\left(-15\right)}$$

Multiply $-4$ times $-15$.

$$z=\frac{-\left(-1\right)±\sqrt{1+60\times 27}}{2\left(-15\right)}$$

Multiply $60$ times $27$.

$$z=\frac{-\left(-1\right)±\sqrt{1+1620}}{2\left(-15\right)}$$

Add $1$ to $1620$.

$$z=\frac{-\left(-1\right)±\sqrt{1621}}{2\left(-15\right)}$$

The opposite of $-1$ is $1$.

$$z=\frac{1±\sqrt{1621}}{2\left(-15\right)}$$

Multiply $2$ times $-15$.

$$z=\frac{1±\sqrt{1621}}{-30}$$

Now solve the equation $z=\frac{1±\sqrt{1621}}{-30}$ when $±$ is plus. Add $1$ to $\sqrt{1621}$.

$$z=\frac{\sqrt{1621}+1}{-30}$$

Divide $1+\sqrt{1621}$ by $-30$.

$$z=\frac{-\sqrt{1621}-1}{30}$$

Now solve the equation $z=\frac{1±\sqrt{1621}}{-30}$ when $±$ is minus. Subtract $\sqrt{1621}$ from $1$.

$$z=\frac{1-\sqrt{1621}}{-30}$$

Divide $1-\sqrt{1621}$ by $-30$.

$$z=\frac{\sqrt{1621}-1}{30}$$

The equation is now solved.

$$z=\frac{-\sqrt{1621}-1}{30}$$ $$z=\frac{\sqrt{1621}-1}{30}$$

Steps for Completing the Square

Variable $z$ cannot be equal to any of the values $-\frac{7}{4},-\frac{2}{9}$ since division by zero is not defined. Multiply both sides of the equation by $\left(4z+7\right)\left(9z+2\right)$, the least common multiple of $9z+2,4z+7$.

$$\left(4z+7\right)\left(3z+5\right)=\left(9z+2\right)\left(3z+4\right)$$

Use the distributive property to multiply $4z+7$ by $3z+5$ and combine like terms.

$$12z^{2}+41z+35=\left(9z+2\right)\left(3z+4\right)$$

Use the distributive property to multiply $9z+2$ by $3z+4$ and combine like terms.

$$12z^{2}+41z+35=27z^{2}+42z+8$$

Subtract $27z^{2}$ from both sides.

$$12z^{2}+41z+35-27z^{2}=42z+8$$

Combine $12z^{2}$ and $-27z^{2}$ to get $-15z^{2}$.

$$-15z^{2}+41z+35=42z+8$$

Subtract $42z$ from both sides.

$$-15z^{2}+41z+35-42z=8$$

Combine $41z$ and $-42z$ to get $-z$.

$$-15z^{2}-z+35=8$$

Subtract $35$ from both sides.

$$-15z^{2}-z=8-35$$

Subtract $35$ from $8$ to get $-27$.

$$-15z^{2}-z=-27$$

Divide both sides by $-15$.

$$\frac{-15z^{2}-z}{-15}=-\frac{27}{-15}$$

Dividing by $-15$ undoes the multiplication by $-15$.

$$z^{2}+\left(-\frac{1}{-15}\right)z=-\frac{27}{-15}$$

Divide $-1$ by $-15$.

$$z^{2}+\frac{1}{15}z=-\frac{27}{-15}$$

Reduce the fraction $\frac{-27}{-15}$ to lowest terms by extracting and canceling out $3$.

$$z^{2}+\frac{1}{15}z=\frac{9}{5}$$

Divide $\frac{1}{15}$, the coefficient of the $x$ term, by $2$ to get $\frac{1}{30}$. Then add the square of $\frac{1}{30}$ to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$z^{2}+\frac{1}{15}z+\left(\frac{1}{30}\right)^{2}=\frac{9}{5}+\left(\frac{1}{30}\right)^{2}$$

Square $\frac{1}{30}$ by squaring both the numerator and the denominator of the fraction.

$$z^{2}+\frac{1}{15}z+\frac{1}{900}=\frac{9}{5}+\frac{1}{900}$$

Add $\frac{9}{5}$ to $\frac{1}{900}$ by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$z^{2}+\frac{1}{15}z+\frac{1}{900}=\frac{1621}{900}$$

Factor $z^{2}+\frac{1}{15}z+\frac{1}{900}$. In general, when $x^{2}+bx+c$ is a perfect square, it can always be factored as $\left(x+\frac{b}{2}\right)^{2}$.

$$\left(z+\frac{1}{30}\right)^{2}=\frac{1621}{900}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(z+\frac{1}{30}\right)^{2}}=\sqrt{\frac{1621}{900}}$$

Simplify.

$$z+\frac{1}{30}=\frac{\sqrt{1621}}{30}$$ $$z+\frac{1}{30}=-\frac{\sqrt{1621}}{30}$$

Subtract $\frac{1}{30}$ from both sides of the equation.

$$z=\frac{\sqrt{1621}-1}{30}$$ $$z=\frac{-\sqrt{1621}-1}{30}$$