Basic Math Examples

$4 t + \frac{1}{3} \cdot (10 t + 18) + 6 t + 6 = 7 t + 5 + \frac{1}{2} \cdot (18 t + 1)$

Step 1

Simplify $4 t + \frac{1}{3} \cdot (10 t + 18) + 6 t + 6$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$4 t + \frac{1}{3} (10 t) + \frac{1}{3} \cdot 18 + 6 t + 6 = 7 t + 5 + \frac{1}{2} \cdot (18 t + 1)$

Step 1.1.2

Multiply $\frac{1}{3} (10 t)$ .

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Step 1.1.2.1

Combine $10$ and $\frac{1}{3}$ .

$4 t + \frac{10}{3} t + \frac{1}{3} \cdot 18 + 6 t + 6 = 7 t + 5 + \frac{1}{2} \cdot (18 t + 1)$

Step 1.1.2.2

Combine $\frac{10}{3}$ and $t$ .

$4 t + \frac{10 t}{3} + \frac{1}{3} \cdot 18 + 6 t + 6 = 7 t + 5 + \frac{1}{2} \cdot (18 t + 1)$

Step 1.1.3

Cancel the common factor of $3$ .

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Step 1.1.3.1

Factor $3$ out of $18$ .

$4 t + \frac{10 t}{3} + \frac{1}{3} \cdot (3 (6)) + 6 t + 6 = 7 t + 5 + \frac{1}{2} \cdot (18 t + 1)$

Step 1.1.3.2

Cancel the common factor.

$4 t + \frac{10 t}{3} + \frac{1}{3} \cdot (3 \cdot 6) + 6 t + 6 = 7 t + 5 + \frac{1}{2} \cdot (18 t + 1)$

Step 1.1.3.3

Rewrite the expression.

$4 t + \frac{10 t}{3} + 6 + 6 t + 6 = 7 t + 5 + \frac{1}{2} \cdot (18 t + 1)$

Step 1.2

To write $4 t$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$4 t \cdot \frac{3}{3} + \frac{10 t}{3} + 6 + 6 t + 6 = 7 t + 5 + \frac{1}{2} \cdot (18 t + 1)$

Step 1.3

Simplify terms.

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Step 1.3.1

Combine $4 t$ and $\frac{3}{3}$ .

$\frac{4 t \cdot 3}{3} + \frac{10 t}{3} + 6 + 6 t + 6 = 7 t + 5 + \frac{1}{2} \cdot (18 t + 1)$

Step 1.3.2

Combine the numerators over the common denominator.

$\frac{4 t \cdot 3 + 10 t}{3} + 6 + 6 t + 6 = 7 t + 5 + \frac{1}{2} \cdot (18 t + 1)$

Step 1.4

Simplify the numerator.

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Step 1.4.1

Factor $2 t$ out of $4 t \cdot 3 + 10 t$ .

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Step 1.4.1.1

Factor $2 t$ out of $4 t \cdot 3$ .

$\frac{2 t (2 \cdot 3) + 10 t}{3} + 6 + 6 t + 6 = 7 t + 5 + \frac{1}{2} \cdot (18 t + 1)$

Step 1.4.1.2

Factor $2 t$ out of $10 t$ .

$\frac{2 t (2 \cdot 3) + 2 t (5)}{3} + 6 + 6 t + 6 = 7 t + 5 + \frac{1}{2} \cdot (18 t + 1)$

Step 1.4.1.3

Factor $2 t$ out of $2 t (2 \cdot 3) + 2 t (5)$ .

$\frac{2 t (2 \cdot 3 + 5)}{3} + 6 + 6 t + 6 = 7 t + 5 + \frac{1}{2} \cdot (18 t + 1)$

Step 1.4.2

Multiply $2$ by $3$ .

$\frac{2 t (6 + 5)}{3} + 6 + 6 t + 6 = 7 t + 5 + \frac{1}{2} \cdot (18 t + 1)$

Step 1.4.3

Add $6$ and $5$ .

$\frac{2 t \cdot 11}{3} + 6 + 6 t + 6 = 7 t + 5 + \frac{1}{2} \cdot (18 t + 1)$

Step 1.4.4

Multiply $11$ by $2$ .

$\frac{22 t}{3} + 6 + 6 t + 6 = 7 t + 5 + \frac{1}{2} \cdot (18 t + 1)$

Step 1.5

To write $6 t$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{22 t}{3} + 6 t \cdot \frac{3}{3} + 6 + 6 = 7 t + 5 + \frac{1}{2} \cdot (18 t + 1)$

Step 1.6

Combine $6 t$ and $\frac{3}{3}$ .

$\frac{22 t}{3} + \frac{6 t \cdot 3}{3} + 6 + 6 = 7 t + 5 + \frac{1}{2} \cdot (18 t + 1)$

Step 1.7

Combine the numerators over the common denominator.

$\frac{22 t + 6 t \cdot 3}{3} + 6 + 6 = 7 t + 5 + \frac{1}{2} \cdot (18 t + 1)$

Step 1.8

Find the common denominator.

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Step 1.8.1

Write $6$ as a fraction with denominator $1$ .

$\frac{22 t + 6 t \cdot 3}{3} + \frac{6}{1} + 6 = 7 t + 5 + \frac{1}{2} \cdot (18 t + 1)$

Step 1.8.2

Multiply $\frac{6}{1}$ by $\frac{3}{3}$ .

$\frac{22 t + 6 t \cdot 3}{3} + \frac{6}{1} \cdot \frac{3}{3} + 6 = 7 t + 5 + \frac{1}{2} \cdot (18 t + 1)$

Step 1.8.3

Multiply $\frac{6}{1}$ by $\frac{3}{3}$ .

$\frac{22 t + 6 t \cdot 3}{3} + \frac{6 \cdot 3}{3} + 6 = 7 t + 5 + \frac{1}{2} \cdot (18 t + 1)$

Step 1.8.4

Write $6$ as a fraction with denominator $1$ .

$\frac{22 t + 6 t \cdot 3}{3} + \frac{6 \cdot 3}{3} + \frac{6}{1} = 7 t + 5 + \frac{1}{2} \cdot (18 t + 1)$

Step 1.8.5

Multiply $\frac{6}{1}$ by $\frac{3}{3}$ .

$\frac{22 t + 6 t \cdot 3}{3} + \frac{6 \cdot 3}{3} + \frac{6}{1} \cdot \frac{3}{3} = 7 t + 5 + \frac{1}{2} \cdot (18 t + 1)$

Step 1.8.6

Multiply $\frac{6}{1}$ by $\frac{3}{3}$ .

$\frac{22 t + 6 t \cdot 3}{3} + \frac{6 \cdot 3}{3} + \frac{6 \cdot 3}{3} = 7 t + 5 + \frac{1}{2} \cdot (18 t + 1)$

Step 1.9

Combine the numerators over the common denominator.

$\frac{22 t + 6 t \cdot 3 + 6 \cdot 3 + 6 \cdot 3}{3} = 7 t + 5 + \frac{1}{2} \cdot (18 t + 1)$

Step 1.10

Simplify each term.

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Step 1.10.1

Multiply $3$ by $6$ .

$\frac{22 t + 18 t + 6 \cdot 3 + 6 \cdot 3}{3} = 7 t + 5 + \frac{1}{2} \cdot (18 t + 1)$

Step 1.10.2

Multiply $6$ by $3$ .

$\frac{22 t + 18 t + 18 + 6 \cdot 3}{3} = 7 t + 5 + \frac{1}{2} \cdot (18 t + 1)$

Step 1.10.3

Multiply $6$ by $3$ .

$\frac{22 t + 18 t + 18 + 18}{3} = 7 t + 5 + \frac{1}{2} \cdot (18 t + 1)$

Step 1.11

Simplify terms.

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Step 1.11.1

Add $22 t$ and $18 t$ .

$\frac{40 t + 18 + 18}{3} = 7 t + 5 + \frac{1}{2} \cdot (18 t + 1)$

Step 1.11.2

Add $18$ and $18$ .

$\frac{40 t + 36}{3} = 7 t + 5 + \frac{1}{2} \cdot (18 t + 1)$

Step 1.11.3

Factor $4$ out of $40 t + 36$ .

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Step 1.11.3.1

Factor $4$ out of $40 t$ .

$\frac{4 (10 t) + 36}{3} = 7 t + 5 + \frac{1}{2} \cdot (18 t + 1)$

Step 1.11.3.2

Factor $4$ out of $36$ .

$\frac{4 (10 t) + 4 (9)}{3} = 7 t + 5 + \frac{1}{2} \cdot (18 t + 1)$

Step 1.11.3.3

Factor $4$ out of $4 (10 t) + 4 (9)$ .

$\frac{4 (10 t + 9)}{3} = 7 t + 5 + \frac{1}{2} \cdot (18 t + 1)$

Step 2

Simplify $7 t + 5 + \frac{1}{2} \cdot (18 t + 1)$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Apply the distributive property.

$\frac{4 (10 t + 9)}{3} = 7 t + 5 + \frac{1}{2} (18 t) + \frac{1}{2} \cdot 1$

Step 2.1.2

Cancel the common factor of $2$ .

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Step 2.1.2.1

Factor $2$ out of $18 t$ .

$\frac{4 (10 t + 9)}{3} = 7 t + 5 + \frac{1}{2} (2 (9 t)) + \frac{1}{2} \cdot 1$

Step 2.1.2.2

Cancel the common factor.

$\frac{4 (10 t + 9)}{3} = 7 t + 5 + \frac{1}{2} (2 (9 t)) + \frac{1}{2} \cdot 1$

Step 2.1.2.3

Rewrite the expression.

$\frac{4 (10 t + 9)}{3} = 7 t + 5 + 9 t + \frac{1}{2} \cdot 1$

Step 2.1.3

Multiply $\frac{1}{2}$ by $1$ .

$\frac{4 (10 t + 9)}{3} = 7 t + 5 + 9 t + \frac{1}{2}$

Step 2.2

Add $7 t$ and $9 t$ .

$\frac{4 (10 t + 9)}{3} = 16 t + 5 + \frac{1}{2}$

Step 2.3

To write $5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{4 (10 t + 9)}{3} = 16 t + 5 \cdot \frac{2}{2} + \frac{1}{2}$

Step 2.4

Combine $5$ and $\frac{2}{2}$ .

$\frac{4 (10 t + 9)}{3} = 16 t + \frac{5 \cdot 2}{2} + \frac{1}{2}$

Step 2.5

Combine the numerators over the common denominator.

$\frac{4 (10 t + 9)}{3} = 16 t + \frac{5 \cdot 2 + 1}{2}$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $5$ by $2$ .

$\frac{4 (10 t + 9)}{3} = 16 t + \frac{10 + 1}{2}$

Step 2.6.2

Add $10$ and $1$ .

$\frac{4 (10 t + 9)}{3} = 16 t + \frac{11}{2}$

Step 3

Move all terms containing $t$ to the left side of the equation.

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Step 3.1

Subtract $16 t$ from both sides of the equation.

$\frac{4 (10 t + 9)}{3} - 16 t = \frac{11}{2}$

Step 3.2

To write $- 16 t$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{4 (10 t + 9)}{3} - 16 t \cdot \frac{3}{3} = \frac{11}{2}$

Step 3.3

Combine $- 16 t$ and $\frac{3}{3}$ .

$\frac{4 (10 t + 9)}{3} + \frac{- 16 t \cdot 3}{3} = \frac{11}{2}$

Step 3.4

Combine the numerators over the common denominator.

$\frac{4 (10 t + 9) - 16 t \cdot 3}{3} = \frac{11}{2}$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Factor $4$ out of $4 (10 t + 9) - 16 t \cdot 3$ .

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Step 3.5.1.1

Factor $4$ out of $- 16 t \cdot 3$ .

$\frac{4 (10 t + 9) + 4 (- 4 t \cdot 3)}{3} = \frac{11}{2}$

Step 3.5.1.2

Factor $4$ out of $4 (10 t + 9) + 4 (- 4 t \cdot 3)$ .

$\frac{4 (10 t + 9 - 4 t \cdot 3)}{3} = \frac{11}{2}$

Step 3.5.2

Multiply $3$ by $- 4$ .

$\frac{4 (10 t + 9 - 12 t)}{3} = \frac{11}{2}$

Step 3.5.3

Subtract $12 t$ from $10 t$ .

$\frac{4 (- 2 t + 9)}{3} = \frac{11}{2}$

Step 3.6

Factor $- 1$ out of $- 2 t$ .

$\frac{4 (- (2 t) + 9)}{3} = \frac{11}{2}$

Step 3.7

Rewrite $9$ as $- 1 (- 9)$ .

$\frac{4 (- (2 t) - 1 (- 9))}{3} = \frac{11}{2}$

Step 3.8

Factor $- 1$ out of $- (2 t) - 1 (- 9)$ .

$\frac{4 (- (2 t - 9))}{3} = \frac{11}{2}$

Step 3.9

Rewrite $- (2 t - 9)$ as $- 1 (2 t - 9)$ .

$\frac{4 (- 1 (2 t - 9))}{3} = \frac{11}{2}$

Step 3.10

Move the negative in front of the fraction.

$- \frac{4 (2 t - 9)}{3} = \frac{11}{2}$

Step 4

Multiply both sides of the equation by $- \frac{3}{4}$ .

$- \frac{3}{4} (- \frac{4 (2 t - 9)}{3}) = - \frac{3}{4} \cdot \frac{11}{2}$

Step 5

Simplify both sides of the equation.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Simplify $- \frac{3}{4} (- \frac{4 (2 t - 9)}{3})$ .

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Step 5.1.1.1

Cancel the common factor of $3$ .

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Step 5.1.1.1.1

Move the leading negative in $- \frac{3}{4}$ into the numerator.

$\frac{- 3}{4} (- \frac{4 (2 t - 9)}{3}) = - \frac{3}{4} \cdot \frac{11}{2}$

Step 5.1.1.1.2

Move the leading negative in $- \frac{4 (2 t - 9)}{3}$ into the numerator.

$\frac{- 3}{4} \cdot \frac{- 4 (2 t - 9)}{3} = - \frac{3}{4} \cdot \frac{11}{2}$

Step 5.1.1.1.3

Factor $3$ out of $- 3$ .

$\frac{3 (- 1)}{4} \cdot \frac{- 4 (2 t - 9)}{3} = - \frac{3}{4} \cdot \frac{11}{2}$

Step 5.1.1.1.4

Cancel the common factor.

$\frac{3 \cdot - 1}{4} \cdot \frac{- 4 (2 t - 9)}{3} = - \frac{3}{4} \cdot \frac{11}{2}$

Step 5.1.1.1.5

Rewrite the expression.

$\frac{- 1}{4} (- 4 (2 t - 9)) = - \frac{3}{4} \cdot \frac{11}{2}$

Step 5.1.1.2

Cancel the common factor of $4$ .

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Step 5.1.1.2.1

Factor $4$ out of $- 4 (2 t - 9)$ .

$\frac{- 1}{4} (4 (- (2 t - 9))) = - \frac{3}{4} \cdot \frac{11}{2}$

Step 5.1.1.2.2

Cancel the common factor.

$\frac{- 1}{4} (4 (- (2 t - 9))) = - \frac{3}{4} \cdot \frac{11}{2}$

Step 5.1.1.2.3

Rewrite the expression.

$- - (2 t - 9) = - \frac{3}{4} \cdot \frac{11}{2}$

Step 5.1.1.3

Multiply.

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Step 5.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 (2 t - 9) = - \frac{3}{4} \cdot \frac{11}{2}$

Step 5.1.1.3.2

Multiply $2 t - 9$ by $1$ .

$2 t - 9 = - \frac{3}{4} \cdot \frac{11}{2}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Multiply $- \frac{3}{4} \cdot \frac{11}{2}$ .

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Step 5.2.1.1

Multiply $\frac{11}{2}$ by $\frac{3}{4}$ .

$2 t - 9 = - \frac{11 \cdot 3}{2 \cdot 4}$

Step 5.2.1.2

Multiply $11$ by $3$ .

$2 t - 9 = - \frac{33}{2 \cdot 4}$

Step 5.2.1.3

Multiply $2$ by $4$ .

$2 t - 9 = - \frac{33}{8}$

Step 6

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

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Step 6.1

Add $9$ to both sides of the equation.

$2 t = - \frac{33}{8} + 9$

Step 6.2

To write $9$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$2 t = - \frac{33}{8} + 9 \cdot \frac{8}{8}$

Step 6.3

Combine $9$ and $\frac{8}{8}$ .

$2 t = - \frac{33}{8} + \frac{9 \cdot 8}{8}$

Step 6.4

Combine the numerators over the common denominator.

$2 t = \frac{- 33 + 9 \cdot 8}{8}$

Step 6.5

Simplify the numerator.

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Step 6.5.1

Multiply $9$ by $8$ .

$2 t = \frac{- 33 + 72}{8}$

Step 6.5.2

Add $- 33$ and $72$ .

$2 t = \frac{39}{8}$

Step 7

Divide each term in $2 t = \frac{39}{8}$ by $2$ and simplify.

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Step 7.1

Divide each term in $2 t = \frac{39}{8}$ by $2$ .

$\frac{2 t}{2} = \frac{\frac{39}{8}}{2}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $2$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{2 t}{2} = \frac{\frac{39}{8}}{2}$

Step 7.2.1.2

Divide $t$ by $1$ .

$t = \frac{\frac{39}{8}}{2}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Multiply the numerator by the reciprocal of the denominator.

$t = \frac{39}{8} \cdot \frac{1}{2}$

Step 7.3.2

Multiply $\frac{39}{8} \cdot \frac{1}{2}$ .

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Step 7.3.2.1

Multiply $\frac{39}{8}$ by $\frac{1}{2}$ .

$t = \frac{39}{8 \cdot 2}$

Step 7.3.2.2

Multiply $8$ by $2$ .

$t = \frac{39}{16}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$t = \frac{39}{16}$

Decimal Form:

$t = 2.4375$

Mixed Number Form:

$t = 2 \frac{7}{16}$