Basic Math Examples

$2 y - \frac{716}{105} = \frac{19}{5 y} + \frac{19}{6}$

Step 1

Move all terms not containing $y$ to the right side of the equation.

Tap for more steps...

Step 1.1

Add $\frac{716}{105}$ to both sides of the equation.

$2 y = \frac{19}{5 y} + \frac{19}{6} + \frac{716}{105}$

Step 1.2

To write $\frac{19}{6}$ as a fraction with a common denominator, multiply by $\frac{35}{35}$ .

$2 y = \frac{19}{5 y} + \frac{19}{6} \cdot \frac{35}{35} + \frac{716}{105}$

Step 1.3

To write $\frac{716}{105}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$2 y = \frac{19}{5 y} + \frac{19}{6} \cdot \frac{35}{35} + \frac{716}{105} \cdot \frac{2}{2}$

Step 1.4

Write each expression with a common denominator of $210$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 1.4.1

Multiply $\frac{19}{6}$ by $\frac{35}{35}$ .

$2 y = \frac{19}{5 y} + \frac{19 \cdot 35}{6 \cdot 35} + \frac{716}{105} \cdot \frac{2}{2}$

Step 1.4.2

Multiply $6$ by $35$ .

$2 y = \frac{19}{5 y} + \frac{19 \cdot 35}{210} + \frac{716}{105} \cdot \frac{2}{2}$

Step 1.4.3

Multiply $\frac{716}{105}$ by $\frac{2}{2}$ .

$2 y = \frac{19}{5 y} + \frac{19 \cdot 35}{210} + \frac{716 \cdot 2}{105 \cdot 2}$

Step 1.4.4

Multiply $105$ by $2$ .

$2 y = \frac{19}{5 y} + \frac{19 \cdot 35}{210} + \frac{716 \cdot 2}{210}$

Step 1.5

Combine the numerators over the common denominator.

$2 y = \frac{19}{5 y} + \frac{19 \cdot 35 + 716 \cdot 2}{210}$

Step 1.6

Simplify the numerator.

Tap for more steps...

Step 1.6.1

Multiply $19$ by $35$ .

$2 y = \frac{19}{5 y} + \frac{665 + 716 \cdot 2}{210}$

Step 1.6.2

Multiply $716$ by $2$ .

$2 y = \frac{19}{5 y} + \frac{665 + 1432}{210}$

Step 1.6.3

Add $665$ and $1432$ .

$2 y = \frac{19}{5 y} + \frac{2097}{210}$

Step 1.7

Cancel the common factor of $2097$ and $210$ .

Tap for more steps...

Step 1.7.1

Factor $3$ out of $2097$ .

$2 y = \frac{19}{5 y} + \frac{3 (699)}{210}$

Step 1.7.2

Cancel the common factors.

Tap for more steps...

Step 1.7.2.1

Factor $3$ out of $210$ .

$2 y = \frac{19}{5 y} + \frac{3 \cdot 699}{3 \cdot 70}$

Step 1.7.2.2

Cancel the common factor.

$2 y = \frac{19}{5 y} + \frac{3 \cdot 699}{3 \cdot 70}$

Step 1.7.2.3

Rewrite the expression.

$2 y = \frac{19}{5 y} + \frac{699}{70}$

Step 2

Find the LCD of the terms in the equation.

Tap for more steps...

Step 2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, 5 y, 70$

Step 2.2

Since $1, 5 y, 70$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $1, 5, 70$ then find LCM for the variable part $y^{1}$ .

Step 2.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 2.4

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 2.5

Since $5$ has no factors besides $1$ and $5$ .

$5$ is a prime number

Step 2.6

The prime factors for $70$ are $2 \cdot 5 \cdot 7$ .

Tap for more steps...

Step 2.6.1

$70$ has factors of $2$ and $35$ .

$2 \cdot 35$

Step 2.6.2

$35$ has factors of $5$ and $7$ .

$2 \cdot 5 \cdot 7$

Step 2.7

Multiply $2 \cdot 5 \cdot 7$ .

Tap for more steps...

Step 2.7.1

Multiply $2$ by $5$ .

$10 \cdot 7$

Step 2.7.2

Multiply $10$ by $7$ .

$70$

Step 2.8

The factor for $y^{1}$ is $y$ itself.

$y^{1} = y$

$y$ occurs $1$ time.

Step 2.9

The LCM of $y^{1}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$y$

Step 2.10

The LCM for $1, 5 y, 70$ is the numeric part $70$ multiplied by the variable part.

$70 y$

Step 3

Multiply each term in $2 y = \frac{19}{5 y} + \frac{699}{70}$ by $70 y$ to eliminate the fractions.

Tap for more steps...

Step 3.1

Multiply each term in $2 y = \frac{19}{5 y} + \frac{699}{70}$ by $70 y$ .

$2 y (70 y) = \frac{19}{5 y} (70 y) + \frac{699}{70} (70 y)$

Step 3.2

Simplify the left side.

Tap for more steps...

Step 3.2.1

Rewrite using the commutative property of multiplication.

$2 \cdot 70 y \cdot y = \frac{19}{5 y} (70 y) + \frac{699}{70} (70 y)$

Step 3.2.2

Multiply $y$ by $y$ by adding the exponents.

Tap for more steps...

Step 3.2.2.1

Move $y$ .

$2 \cdot 70 (y \cdot y) = \frac{19}{5 y} (70 y) + \frac{699}{70} (70 y)$

Step 3.2.2.2

Multiply $y$ by $y$ .

$2 \cdot 70 y^{2} = \frac{19}{5 y} (70 y) + \frac{699}{70} (70 y)$

Step 3.2.3

Multiply $2$ by $70$ .

$140 y^{2} = \frac{19}{5 y} (70 y) + \frac{699}{70} (70 y)$

Step 3.3

Simplify the right side.

Tap for more steps...

Step 3.3.1

Simplify each term.

Tap for more steps...

Step 3.3.1.1

Rewrite using the commutative property of multiplication.

$140 y^{2} = 70 \frac{19}{5 y} y + \frac{699}{70} (70 y)$

Step 3.3.1.2

Cancel the common factor of $5$ .

Tap for more steps...

Step 3.3.1.2.1

Factor $5$ out of $70$ .

$140 y^{2} = 5 (14) \frac{19}{5 y} y + \frac{699}{70} (70 y)$

Step 3.3.1.2.2

Factor $5$ out of $5 y$ .

$140 y^{2} = 5 (14) \frac{19}{5 (y)} y + \frac{699}{70} (70 y)$

Step 3.3.1.2.3

Cancel the common factor.

$140 y^{2} = 5 \cdot 14 \frac{19}{5 y} y + \frac{699}{70} (70 y)$

Step 3.3.1.2.4

Rewrite the expression.

$140 y^{2} = 14 \frac{19}{y} y + \frac{699}{70} (70 y)$

Step 3.3.1.3

Combine $14$ and $\frac{19}{y}$ .

$140 y^{2} = \frac{14 \cdot 19}{y} y + \frac{699}{70} (70 y)$

Step 3.3.1.4

Multiply $14$ by $19$ .

$140 y^{2} = \frac{266}{y} y + \frac{699}{70} (70 y)$

Step 3.3.1.5

Cancel the common factor of $y$ .

Tap for more steps...

Step 3.3.1.5.1

Cancel the common factor.

$140 y^{2} = \frac{266}{y} y + \frac{699}{70} (70 y)$

Step 3.3.1.5.2

Rewrite the expression.

$140 y^{2} = 266 + \frac{699}{70} (70 y)$

Step 3.3.1.6

Cancel the common factor of $70$ .

Tap for more steps...

Step 3.3.1.6.1

Factor $70$ out of $70 y$ .

$140 y^{2} = 266 + \frac{699}{70} (70 (y))$

Step 3.3.1.6.2

Cancel the common factor.

$140 y^{2} = 266 + \frac{699}{70} (70 y)$

Step 3.3.1.6.3

Rewrite the expression.

$140 y^{2} = 266 + 699 y$

Step 4

Solve the equation.

Tap for more steps...

Step 4.1

Subtract $699 y$ from both sides of the equation.

$140 y^{2} - 699 y = 266$

Step 4.2

Subtract $266$ from both sides of the equation.

$140 y^{2} - 699 y - 266 = 0$

Step 4.3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4.4

Substitute the values $a = 140$ , $b = - 699$ , and $c = - 266$ into the quadratic formula and solve for $y$ .

$\frac{699 \pm \sqrt{{(- 699)}^{2} - 4 \cdot (140 \cdot - 266)}}{2 \cdot 140}$

Step 4.5

Simplify.

Tap for more steps...

Step 4.5.1

Simplify the numerator.

Tap for more steps...

Step 4.5.1.1

Raise $- 699$ to the power of $2$ .

$y = \frac{699 \pm \sqrt{488601 - 4 \cdot 140 \cdot - 266}}{2 \cdot 140}$

Step 4.5.1.2

Multiply $- 4 \cdot 140 \cdot - 266$ .

Tap for more steps...

Step 4.5.1.2.1

Multiply $- 4$ by $140$ .

$y = \frac{699 \pm \sqrt{488601 - 560 \cdot - 266}}{2 \cdot 140}$

Step 4.5.1.2.2

Multiply $- 560$ by $- 266$ .

$y = \frac{699 \pm \sqrt{488601 + 148960}}{2 \cdot 140}$

Step 4.5.1.3

Add $488601$ and $148960$ .

$y = \frac{699 \pm \sqrt{637561}}{2 \cdot 140}$

Step 4.5.2

Multiply $2$ by $140$ .

$y = \frac{699 \pm \sqrt{637561}}{280}$

Step 4.6

The final answer is the combination of both solutions.

$y = \frac{699 + \sqrt{637561}}{280}, \frac{699 - \sqrt{637561}}{280}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$y = \frac{699 + \sqrt{637561}}{280}, \frac{699 - \sqrt{637561}}{280}$

Decimal Form:

$y = 5.34812203 \dots, - 0.35526489 \dots$