Solve ) 8 + a - (10a)/3 = 47/6 - (5a)/2; (p - 7)/6 | 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

$$8 + a - \frac { 10 a } { 3 } = \frac { 47 } { 6 } - \frac { 5 a } { 2 } \quad ( x ) \frac { p - 7 } { 6 }$$

Solve for p (complex solution)

$\left\{\begin{matrix}p=\frac{35ax+28a-2}{5ax}\text{, }&x\neq 0\text{ and }a\neq 0\\p\in \mathrm{C}\text{, }&a=\frac{1}{14}\text{ and }x=0\end{matrix}\right.$

Steps for Solving Linear Equation

Multiply both sides of the equation by $6$, the least common multiple of $3,6,2$.

$$48+6a-2\times 10a=47-3\times 5ax\times \frac{p-7}{6}$$

Multiply $-2$ and $10$ to get $-20$.

$$48+6a-20a=47-3\times 5ax\times \frac{p-7}{6}$$

Combine $6a$ and $-20a$ to get $-14a$.

$$48-14a=47-3\times 5ax\times \frac{p-7}{6}$$

Multiply $3$ and $5$ to get $15$.

$$48-14a=47-15ax\times \frac{p-7}{6}$$

Express $15\times \frac{p-7}{6}$ as a single fraction.

$$48-14a=47-\frac{15\left(p-7\right)}{6}ax$$

Use the distributive property to multiply $15$ by $p-7$.

$$48-14a=47-\frac{15p-105}{6}ax$$

Divide each term of $15p-105$ by $6$ to get $\frac{5}{2}p-\frac{35}{2}$.

$$48-14a=47-\left(\frac{5}{2}p-\frac{35}{2}\right)ax$$

Use the distributive property to multiply $\frac{5}{2}p-\frac{35}{2}$ by $a$.

$$48-14a=47-\left(\frac{5}{2}pa-\frac{35}{2}a\right)x$$

Use the distributive property to multiply $\frac{5}{2}pa-\frac{35}{2}a$ by $x$.

$$48-14a=47-\left(\frac{5}{2}pax-\frac{35}{2}ax\right)$$

To find the opposite of $\frac{5}{2}pax-\frac{35}{2}ax$, find the opposite of each term.

$$48-14a=47-\frac{5}{2}pax+\frac{35}{2}ax$$

Swap sides so that all variable terms are on the left hand side.

$$47-\frac{5}{2}pax+\frac{35}{2}ax=48-14a$$

Subtract $47$ from both sides.

$$-\frac{5}{2}pax+\frac{35}{2}ax=48-14a-47$$

Subtract $47$ from $48$ to get $1$.

$$-\frac{5}{2}pax+\frac{35}{2}ax=1-14a$$

Subtract $\frac{35}{2}ax$ from both sides.

$$-\frac{5}{2}pax=1-14a-\frac{35}{2}ax$$

The equation is in standard form.

$$\left(-\frac{5ax}{2}\right)p=-\frac{35ax}{2}-14a+1$$

Divide both sides by $-\frac{5}{2}ax$.

$$\frac{\left(-\frac{5ax}{2}\right)p}{-\frac{5ax}{2}}=\frac{-\frac{35ax}{2}-14a+1}{-\frac{5ax}{2}}$$

Dividing by $-\frac{5}{2}ax$ undoes the multiplication by $-\frac{5}{2}ax$.

$$p=\frac{-\frac{35ax}{2}-14a+1}{-\frac{5ax}{2}}$$

Divide $1-14a-\frac{35ax}{2}$ by $-\frac{5}{2}ax$.

$$p=7+\frac{28a-2}{5ax}$$

Solve for a

$a=-\frac{2}{5px-35x-28}$
$p=7\text{ or }x\neq \frac{28}{5\left(p-7\right)}$

Steps for Solving Linear Equation

Multiply both sides of the equation by $6$, the least common multiple of $3,6,2$.

$$48+6a-2\times 10a=47-3\times 5ax\times \frac{p-7}{6}$$

Multiply $-2$ and $10$ to get $-20$.

$$48+6a-20a=47-3\times 5ax\times \frac{p-7}{6}$$

Combine $6a$ and $-20a$ to get $-14a$.

$$48-14a=47-3\times 5ax\times \frac{p-7}{6}$$

Multiply $3$ and $5$ to get $15$.

$$48-14a=47-15ax\times \frac{p-7}{6}$$

Express $15\times \frac{p-7}{6}$ as a single fraction.

$$48-14a=47-\frac{15\left(p-7\right)}{6}ax$$

Use the distributive property to multiply $15$ by $p-7$.

$$48-14a=47-\frac{15p-105}{6}ax$$

Divide each term of $15p-105$ by $6$ to get $\frac{5}{2}p-\frac{35}{2}$.

$$48-14a=47-\left(\frac{5}{2}p-\frac{35}{2}\right)ax$$

Use the distributive property to multiply $\frac{5}{2}p-\frac{35}{2}$ by $a$.

$$48-14a=47-\left(\frac{5}{2}pa-\frac{35}{2}a\right)x$$

Use the distributive property to multiply $\frac{5}{2}pa-\frac{35}{2}a$ by $x$.

$$48-14a=47-\left(\frac{5}{2}pax-\frac{35}{2}ax\right)$$

To find the opposite of $\frac{5}{2}pax-\frac{35}{2}ax$, find the opposite of each term.

$$48-14a=47-\frac{5}{2}pax+\frac{35}{2}ax$$

Add $\frac{5}{2}pax$ to both sides.

$$48-14a+\frac{5}{2}pax=47+\frac{35}{2}ax$$

Subtract $\frac{35}{2}ax$ from both sides.

$$48-14a+\frac{5}{2}pax-\frac{35}{2}ax=47$$

Subtract $48$ from both sides.

$$-14a+\frac{5}{2}pax-\frac{35}{2}ax=47-48$$

Subtract $48$ from $47$ to get $-1$.

$$-14a+\frac{5}{2}pax-\frac{35}{2}ax=-1$$

Combine all terms containing $a$.

$$\left(-14+\frac{5}{2}px-\frac{35}{2}x\right)a=-1$$

The equation is in standard form.

$$\left(\frac{5px}{2}-\frac{35x}{2}-14\right)a=-1$$

Divide both sides by $-14+\frac{5}{2}px-\frac{35}{2}x$.

$$\frac{\left(\frac{5px}{2}-\frac{35x}{2}-14\right)a}{\frac{5px}{2}-\frac{35x}{2}-14}=-\frac{1}{\frac{5px}{2}-\frac{35x}{2}-14}$$

Dividing by $-14+\frac{5}{2}px-\frac{35}{2}x$ undoes the multiplication by $-14+\frac{5}{2}px-\frac{35}{2}x$.

$$a=-\frac{1}{\frac{5px}{2}-\frac{35x}{2}-14}$$

Divide $-1$ by $-14+\frac{5}{2}px-\frac{35}{2}x$.

$$a=-\frac{2}{5px-35x-28}$$

Solve for p

$\left\{\begin{matrix}p=\frac{35ax+28a-2}{5ax}\text{, }&x\neq 0\text{ and }a\neq 0\\p\in \mathrm{R}\text{, }&a=\frac{1}{14}\text{ and }x=0\end{matrix}\right.$

Steps for Solving Linear Equation

Multiply both sides of the equation by $6$, the least common multiple of $3,6,2$.

$$48+6a-2\times 10a=47-3\times 5ax\times \frac{p-7}{6}$$

Multiply $-2$ and $10$ to get $-20$.

$$48+6a-20a=47-3\times 5ax\times \frac{p-7}{6}$$

Combine $6a$ and $-20a$ to get $-14a$.

$$48-14a=47-3\times 5ax\times \frac{p-7}{6}$$

Multiply $3$ and $5$ to get $15$.

$$48-14a=47-15ax\times \frac{p-7}{6}$$

Express $15\times \frac{p-7}{6}$ as a single fraction.

$$48-14a=47-\frac{15\left(p-7\right)}{6}ax$$

Use the distributive property to multiply $15$ by $p-7$.

$$48-14a=47-\frac{15p-105}{6}ax$$

Divide each term of $15p-105$ by $6$ to get $\frac{5}{2}p-\frac{35}{2}$.

$$48-14a=47-\left(\frac{5}{2}p-\frac{35}{2}\right)ax$$

Use the distributive property to multiply $\frac{5}{2}p-\frac{35}{2}$ by $a$.

$$48-14a=47-\left(\frac{5}{2}pa-\frac{35}{2}a\right)x$$

Use the distributive property to multiply $\frac{5}{2}pa-\frac{35}{2}a$ by $x$.

$$48-14a=47-\left(\frac{5}{2}pax-\frac{35}{2}ax\right)$$

To find the opposite of $\frac{5}{2}pax-\frac{35}{2}ax$, find the opposite of each term.

$$48-14a=47-\frac{5}{2}pax+\frac{35}{2}ax$$

Swap sides so that all variable terms are on the left hand side.

$$47-\frac{5}{2}pax+\frac{35}{2}ax=48-14a$$

Subtract $47$ from both sides.

$$-\frac{5}{2}pax+\frac{35}{2}ax=48-14a-47$$

Subtract $47$ from $48$ to get $1$.

$$-\frac{5}{2}pax+\frac{35}{2}ax=1-14a$$

Subtract $\frac{35}{2}ax$ from both sides.

$$-\frac{5}{2}pax=1-14a-\frac{35}{2}ax$$

The equation is in standard form.

$$\left(-\frac{5ax}{2}\right)p=-\frac{35ax}{2}-14a+1$$

Divide both sides by $-\frac{5}{2}ax$.

$$\frac{\left(-\frac{5ax}{2}\right)p}{-\frac{5ax}{2}}=\frac{-\frac{35ax}{2}-14a+1}{-\frac{5ax}{2}}$$

Dividing by $-\frac{5}{2}ax$ undoes the multiplication by $-\frac{5}{2}ax$.

$$p=\frac{-\frac{35ax}{2}-14a+1}{-\frac{5ax}{2}}$$

Divide $1-14a-\frac{35ax}{2}$ by $-\frac{5}{2}ax$.

$$p=7+\frac{28a-2}{5ax}$$