Pre-Algebra Examples

$\frac{6.25 x + 3.25 y}{160} = 5.13$

Step 1

The standard form of a linear equation is $A x + B y = C$ .

Step 2

Change $6.25$ into a fraction.

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Step 2.1

Multiply by $\frac{100}{100}$ to remove the decimal.

$\frac{\frac{100 \cdot 6.25}{100} x + 3.25 y}{160} = 5.13$

Step 2.2

Multiply $100$ by $6.25$ .

$\frac{\frac{625}{100} x + 3.25 y}{160} = 5.13$

Step 2.3

Cancel the common factor of $625$ and $100$ .

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Step 2.3.1

Factor $25$ out of $625$ .

$\frac{\frac{25 (25)}{100} x + 3.25 y}{160} = 5.13$

Step 2.3.2

Cancel the common factors.

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Step 2.3.2.1

Factor $25$ out of $100$ .

$\frac{\frac{25 \cdot 25}{25 \cdot 4} x + 3.25 y}{160} = 5.13$

Step 2.3.2.2

Cancel the common factor.

$\frac{\frac{25 \cdot 25}{25 \cdot 4} x + 3.25 y}{160} = 5.13$

Step 2.3.2.3

Rewrite the expression.

$\frac{\frac{25}{4} x + 3.25 y}{160} = 5.13$

Step 3

Change $3.25$ into a fraction.

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Step 3.1

Multiply by $\frac{100}{100}$ to remove the decimal.

$\frac{\frac{25}{4} x + \frac{100 \cdot 3.25}{100} y}{160} = 5.13$

Step 3.2

Multiply $100$ by $3.25$ .

$\frac{\frac{25}{4} x + \frac{325}{100} y}{160} = 5.13$

Step 3.3

Cancel the common factor of $325$ and $100$ .

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Step 3.3.1

Factor $25$ out of $325$ .

$\frac{\frac{25}{4} x + \frac{25 (13)}{100} y}{160} = 5.13$

Step 3.3.2

Cancel the common factors.

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Step 3.3.2.1

Factor $25$ out of $100$ .

$\frac{\frac{25}{4} x + \frac{25 \cdot 13}{25 \cdot 4} y}{160} = 5.13$

Step 3.3.2.2

Cancel the common factor.

$\frac{\frac{25}{4} x + \frac{25 \cdot 13}{25 \cdot 4} y}{160} = 5.13$

Step 3.3.2.3

Rewrite the expression.

$\frac{\frac{25}{4} x + \frac{13}{4} y}{160} = 5.13$

Step 4

Change $5.13$ into a fraction.

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Step 4.1

Multiply by $\frac{100}{100}$ to remove the decimal.

$\frac{\frac{25}{4} x + \frac{13}{4} y}{160} = \frac{100 \cdot 5.13}{100}$

Step 4.2

Multiply $100$ by $5.13$ .

$\frac{\frac{25}{4} x + \frac{13}{4} y}{160} = \frac{513}{100}$

Step 4.3

Cancel the common factor of $513$ and $100$ .

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Step 4.3.1

Rewrite $513$ as $1 (513)$ .

$\frac{\frac{25}{4} x + \frac{13}{4} y}{160} = \frac{1 (513)}{100}$

Step 4.3.2

Cancel the common factors.

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Step 4.3.2.1

Rewrite $100$ as $1 (100)$ .

$\frac{\frac{25}{4} x + \frac{13}{4} y}{160} = \frac{1 \cdot 513}{1 \cdot 100}$

Step 4.3.2.2

Cancel the common factor.

$\frac{\frac{25}{4} x + \frac{13}{4} y}{160} = \frac{1 \cdot 513}{1 \cdot 100}$

Step 4.3.2.3

Rewrite the expression.

$\frac{\frac{25}{4} x + \frac{13}{4} y}{160} = \frac{513}{100}$

Step 5

Find the LCD of $\frac{25}{4}, \frac{13}{4}, \frac{\frac{25}{4} x + \frac{13}{4} y}{160},$ and $\frac{513}{100}$ .

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Combine $\frac{25}{4}$ and $x$ .

$\frac{25}{4} + \frac{13}{4} + \frac{\frac{25 x}{4} + \frac{13}{4} y}{160} + \frac{513}{100}$

Step 5.1.2

Combine $\frac{13}{4}$ and $y$ .

$\frac{25}{4} + \frac{13}{4} + \frac{\frac{25 x}{4} + \frac{13 y}{4}}{160} + \frac{513}{100}$

Step 5.1.3

Combine the numerators over the common denominator.

$\frac{25}{4} + \frac{13}{4} + \frac{\frac{25 x + 13 y}{4}}{160} + \frac{513}{100}$

Step 5.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{25}{4} + \frac{13}{4} + \frac{25 x + 13 y}{4} \cdot \frac{1}{160} + \frac{513}{100}$

Step 5.3

Multiply $\frac{25 x + 13 y}{4} \cdot \frac{1}{160}$ .

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Step 5.3.1

Multiply $\frac{25 x + 13 y}{4}$ by $\frac{1}{160}$ .

$\frac{25}{4} + \frac{13}{4} + \frac{25 x + 13 y}{4 \cdot 160} + \frac{513}{100}$

Step 5.3.2

Multiply $4$ by $160$ .

$\frac{25}{4} + \frac{13}{4} + \frac{25 x + 13 y}{640} + \frac{513}{100}$

Step 5.4

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

$4, 4, 640, 100$

Step 5.5

Since $4, 4, 640, 100$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $4, 4, 640, 100$ then find LCM for the variable part .

Step 5.6

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 5.7

$4$ has factors of $2$ and $2$ .

$2 \cdot 2$

Step 5.8

The prime factors for $640$ are $2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 5$ .

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Step 5.8.1

$640$ has factors of $2$ and $320$ .

$2 \cdot 320$

Step 5.8.2

$320$ has factors of $2$ and $160$ .

$2 \cdot 2 \cdot 160$

Step 5.8.3

$160$ has factors of $2$ and $80$ .

$2 \cdot 2 \cdot 2 \cdot 80$

Step 5.8.4

$80$ has factors of $2$ and $40$ .

$2 \cdot 2 \cdot 2 \cdot 2 \cdot 40$

Step 5.8.5

$40$ has factors of $2$ and $20$ .

$2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 20$

Step 5.8.6

$20$ has factors of $2$ and $10$ .

$2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 10$

Step 5.8.7

$10$ has factors of $2$ and $5$ .

$2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 5$

Step 5.9

The prime factors for $100$ are $2 \cdot 2 \cdot 5 \cdot 5$ .

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Step 5.9.1

$100$ has factors of $2$ and $50$ .

$2 \cdot 50$

Step 5.9.2

$50$ has factors of $2$ and $25$ .

$2 \cdot 2 \cdot 25$

Step 5.9.3

$25$ has factors of $5$ and $5$ .

$2 \cdot 2 \cdot 5 \cdot 5$

Step 5.10

The LCM of $4, 4, 640, 100$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 5 \cdot 5$

Step 5.11

Multiply $2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 5 \cdot 5$ .

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Step 5.11.1

Multiply $2$ by $2$ .

$4 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 5 \cdot 5$

Step 5.11.2

Multiply $4$ by $2$ .

$8 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 5 \cdot 5$

Step 5.11.3

Multiply $8$ by $2$ .

$16 \cdot 2 \cdot 2 \cdot 2 \cdot 5 \cdot 5$

Step 5.11.4

Multiply $16$ by $2$ .

$32 \cdot 2 \cdot 2 \cdot 5 \cdot 5$

Step 5.11.5

Multiply $32$ by $2$ .

$64 \cdot 2 \cdot 5 \cdot 5$

Step 5.11.6

Multiply $64$ by $2$ .

$128 \cdot 5 \cdot 5$

Step 5.11.7

Multiply $128$ by $5$ .

$640 \cdot 5$

Step 5.11.8

Multiply $640$ by $5$ .

$3200$

Step 6

Multiply both sides by $3200$ .

$3200 \frac{\frac{25}{4} x + \frac{13}{4} y}{160} = 3200 (\frac{513}{100})$

Step 7

Simplify the left side.

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Step 7.1

Simplify $3200 \frac{\frac{25}{4} x + \frac{13}{4} y}{160}$ .

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Step 7.1.1

Cancel the common factor of $160$ .

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Step 7.1.1.1

Factor $160$ out of $3200$ .

$160 (20) \frac{\frac{25}{4} x + \frac{13}{4} y}{160} = 3200 (\frac{513}{100})$

Step 7.1.1.2

Cancel the common factor.

$160 \cdot 20 \frac{\frac{25}{4} x + \frac{13}{4} y}{160} = 3200 (\frac{513}{100})$

Step 7.1.1.3

Rewrite the expression.

$20 (\frac{25}{4} x + \frac{13}{4} y) = 3200 (\frac{513}{100})$

Step 7.1.2

Combine $\frac{25}{4}$ and $x$ .

$20 (\frac{25 x}{4} + \frac{13}{4} y) = 3200 (\frac{513}{100})$

Step 7.1.3

Combine $\frac{13}{4}$ and $y$ .

$20 (\frac{25 x}{4} + \frac{13 y}{4}) = 3200 (\frac{513}{100})$

Step 7.1.4

Apply the distributive property.

$20 \frac{25 x}{4} + 20 \frac{13 y}{4} = 3200 (\frac{513}{100})$

Step 7.1.5

Cancel the common factor of $4$ .

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Step 7.1.5.1

Factor $4$ out of $20$ .

$4 (5) \frac{25 x}{4} + 20 \frac{13 y}{4} = 3200 (\frac{513}{100})$

Step 7.1.5.2

Cancel the common factor.

$4 \cdot 5 \frac{25 x}{4} + 20 \frac{13 y}{4} = 3200 (\frac{513}{100})$

Step 7.1.5.3

Rewrite the expression.

$5 (25 x) + 20 \frac{13 y}{4} = 3200 (\frac{513}{100})$

Step 7.1.6

Multiply $25$ by $5$ .

$125 x + 20 \frac{13 y}{4} = 3200 (\frac{513}{100})$

Step 7.1.7

Cancel the common factor of $4$ .

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Step 7.1.7.1

Factor $4$ out of $20$ .

$125 x + 4 (5) \frac{13 y}{4} = 3200 (\frac{513}{100})$

Step 7.1.7.2

Cancel the common factor.

$125 x + 4 \cdot 5 \frac{13 y}{4} = 3200 (\frac{513}{100})$

Step 7.1.7.3

Rewrite the expression.

$125 x + 5 (13 y) = 3200 (\frac{513}{100})$

Step 7.1.8

Multiply $13$ by $5$ .

$125 x + 65 y = 3200 (\frac{513}{100})$

Step 8

Simplify the right side.

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Step 8.1

Simplify $3200 (\frac{513}{100})$ .

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Step 8.1.1

Cancel the common factor of $100$ .

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Step 8.1.1.1

Factor $100$ out of $3200$ .

$125 x + 65 y = 100 (32) \frac{513}{100}$

Step 8.1.1.2

Cancel the common factor.

$125 x + 65 y = 100 \cdot 32 \frac{513}{100}$

Step 8.1.1.3

Rewrite the expression.

$125 x + 65 y = 32 \cdot 513$

Step 8.1.2

Multiply $32$ by $513$ .

$125 x + 65 y = 16416$

Step 9