Math / Algebra

QuestionPerform the indicated operation. $\frac{8 a}{5 a-5}-\frac{a-7}{a-1}$ $\frac{8 a}{5 a-5}-\frac{a-7}{a-1}=$ $\square$ (Simplify your answer.)

Studdy Solution

STEP 1

What is this asking? We need to subtract two fractions with variables and simplify the result! Watch out! Don't forget to find a common denominator before subtracting fractions.
Also, be careful with distributing the negative sign when subtracting the second fraction.

STEP 2

1. Factor the denominators
2. Find the least common denominator
3. Rewrite fractions with the common denominator
4. Subtract the fractions
5. Simplify the result

STEP 3

Let's factor the denominators of both fractions to see what we're working with.
The first denominator is $5a - 5$ .
We can factor out a 5, so $5a - 5 = 5(a - 1)$ .
The second denominator is $a - 1$ , which is already factored.

STEP 4

Now, let's find the least common denominator (LCD).
The denominators are $5(a - 1)$ and $(a - 1)$ .
The LCD is the smallest expression that both denominators divide into evenly.
In this case, it's $5(a - 1)$ .

STEP 5

The first fraction already has the LCD as its denominator: $\frac{8a}{5(a - 1)}$ .
Awesome!

STEP 6

The second fraction needs to be rewritten with the LCD.
We multiply the numerator and denominator of $\frac{a - 7}{a - 1}$ by 5 to get it to the LCD.
This gives us $\frac{5 \cdot (a - 7)}{5(a - 1)} = \frac{5a - 35}{5(a - 1)}$ .
Remember, we're multiplying by $\frac{5}{5}$ , which is equal to one, so we're not changing the value of the fraction, just its appearance.

STEP 7

Now we can subtract the fractions since they have a common denominator: $\frac{8a}{5(a - 1)} - \frac{5a - 35}{5(a - 1)} = \frac{8a - (5a - 35)}{5(a - 1)}$

STEP 8

Distribute the negative sign in the numerator: $\frac{8a - 5a + 35}{5(a - 1)}$

STEP 9

Combine like terms in the numerator: $\frac{3a + 35}{5(a - 1)}$

STEP 10

We check if the numerator can be factored further.
We can factor out a 3, but that won't cancel with anything in the denominator, so we can say $\frac{3a + 35}{5(a - 1)}$ is our simplified answer!

STEP 11

$\frac{3a + 35}{5(a - 1)}$

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QuestionPerform the indicated operation. $\frac{8 a}{5 a-5}-\frac{a-7}{a-1}$ $\frac{8 a}{5 a-5}-\frac{a-7}{a-1}=$ $\square$ (Simplify your answer.)

Studdy Solution

STEP 1

What is this asking? We need to subtract two fractions with variables and simplify the result! Watch out! Don't forget to find a common denominator before subtracting fractions.
Also, be careful with distributing the negative sign when subtracting the second fraction.

STEP 2

1. Factor the denominators
2. Find the least common denominator
3. Rewrite fractions with the common denominator
4. Subtract the fractions
5. Simplify the result

STEP 3

Let's factor the denominators of both fractions to see what we're working with.
The first denominator is $5a - 5$ .
We can factor out a 5, so $5a - 5 = 5(a - 1)$ .
The second denominator is $a - 1$ , which is already factored.

STEP 4

Now, let's find the least common denominator (LCD).
The denominators are $5(a - 1)$ and $(a - 1)$ .
The LCD is the smallest expression that both denominators divide into evenly.
In this case, it's $5(a - 1)$ .

STEP 5

The first fraction already has the LCD as its denominator: $\frac{8a}{5(a - 1)}$ .
Awesome!

STEP 6

STEP 7

Now we can subtract the fractions since they have a common denominator: $\frac{8a}{5(a - 1)} - \frac{5a - 35}{5(a - 1)} = \frac{8a - (5a - 35)}{5(a - 1)}$

STEP 8

Distribute the negative sign in the numerator: $\frac{8a - 5a + 35}{5(a - 1)}$

STEP 9

Combine like terms in the numerator: $\frac{3a + 35}{5(a - 1)}$

STEP 10

We check if the numerator can be factored further.
We can factor out a 3, but that won't cancel with anything in the denominator, so we can say $\frac{3a + 35}{5(a - 1)}$ is our simplified answer!

STEP 11

$\frac{3a + 35}{5(a - 1)}$

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QuestionPerform the indicated operation. 8a5a−5−a−7a−1\frac{8 a}{5 a-5}-\frac{a-7}{a-1}5a−58a​−a−1a−7​ 8a5a−5−a−7a−1=\frac{8 a}{5 a-5}-\frac{a-7}{a-1}=5a−58a​−a−1a−7​= □\square□ (Simplify your answer.)

Studdy Solution

STEP 1

What is this asking? We need to subtract two fractions with variables and simplify the result! Watch out! Don't forget to find a common denominator before subtracting fractions.Also, be careful with distributing the negative sign when subtracting the second fraction.

STEP 2

1. Factor the denominators2. Find the least common denominator3. Rewrite fractions with the common denominator4. Subtract the fractions5. Simplify the result

STEP 3

Let's **factor** the denominators of both fractions to see what we're working with.The first denominator is 5a−55a - 55a−5.We can factor out a **5**, so 5a−5=5(a−1)5a - 5 = 5(a - 1)5a−5=5(a−1).The second denominator is a−1a - 1a−1, which is already factored.

STEP 4

Now, let's find the **least common denominator (LCD)**.The denominators are 5(a−1)5(a - 1)5(a−1) and (a−1)(a - 1)(a−1).The LCD is the smallest expression that both denominators divide into evenly.In this case, it's 5(a−1)5(a - 1)5(a−1).

STEP 5

The first fraction already has the LCD as its denominator: 8a5(a−1)\frac{8a}{5(a - 1)}5(a−1)8a​.Awesome!

STEP 6

STEP 7

Now we can subtract the fractions since they have a common denominator: 8a5(a−1)−5a−355(a−1)=8a−(5a−35)5(a−1) \frac{8a}{5(a - 1)} - \frac{5a - 35}{5(a - 1)} = \frac{8a - (5a - 35)}{5(a - 1)} 5(a−1)8a​−5(a−1)5a−35​=5(a−1)8a−(5a−35)​

STEP 8

Distribute the negative sign in the numerator: 8a−5a+355(a−1) \frac{8a - 5a + 35}{5(a - 1)} 5(a−1)8a−5a+35​

STEP 9

Combine like terms in the numerator: 3a+355(a−1) \frac{3a + 35}{5(a - 1)} 5(a−1)3a+35​

STEP 10

We check if the numerator can be factored further.We can factor out a 3, but that won't cancel with anything in the denominator, so we can say 3a+355(a−1)\frac{3a + 35}{5(a - 1)}5(a−1)3a+35​ is our **simplified answer**!

STEP 11

3a+355(a−1)\frac{3a + 35}{5(a - 1)}5(a−1)3a+35​

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Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionPerform the indicated operation. $\frac{8 a}{5 a-5}-\frac{a-7}{a-1}$ $\frac{8 a}{5 a-5}-\frac{a-7}{a-1}=$ $\square$ (Simplify your answer.)

What is this asking? We need to subtract two fractions with variables and simplify the result! Watch out! Don't forget to find a common denominator before subtracting fractions.
Also, be careful with distributing the negative sign when subtracting the second fraction.

1. Factor the denominators
2. Find the least common denominator
3. Rewrite fractions with the common denominator
4. Subtract the fractions
5. Simplify the result

Let's factor the denominators of both fractions to see what we're working with.
The first denominator is $5a - 5$ .
We can factor out a 5, so $5a - 5 = 5(a - 1)$ .
The second denominator is $a - 1$ , which is already factored.

Now, let's find the least common denominator (LCD).
The denominators are $5(a - 1)$ and $(a - 1)$ .
The LCD is the smallest expression that both denominators divide into evenly.
In this case, it's $5(a - 1)$ .

The first fraction already has the LCD as its denominator: $\frac{8a}{5(a - 1)}$ .
Awesome!

Now we can subtract the fractions since they have a common denominator: $\frac{8a}{5(a - 1)} - \frac{5a - 35}{5(a - 1)} = \frac{8a - (5a - 35)}{5(a - 1)}$

Distribute the negative sign in the numerator: $\frac{8a - 5a + 35}{5(a - 1)}$

Combine like terms in the numerator: $\frac{3a + 35}{5(a - 1)}$

We check if the numerator can be factored further.
We can factor out a 3, but that won't cancel with anything in the denominator, so we can say $\frac{3a + 35}{5(a - 1)}$ is our simplified answer!

$\frac{3a + 35}{5(a - 1)}$