Math / Algebra

QuestionSolve each equation. Remember to check for extraneous solutions.
11) $\frac{a-6}{6a^2} = \frac{1}{6a^2} + \frac{a+5}{3a^2}$

Studdy Solution

STEP 1

What is this asking? We need to find the value(s) of $a$ that make this equation true, and make sure those values don't cause any problems, like dividing by zero! Watch out! Be super careful when dealing with variables in the denominator; some values might make the denominator zero, which is a big no-no!

STEP 2

1. Find Common Denominator
2. Combine Fractions
3. Solve for a
4. Check for Extraneous Solutions

STEP 3

Alright, let's get this party started!
We've got fractions with different denominators, so our first mission is to find a common denominator.
Looking at $6a^2$ , $6a^2$ , and $3a^2$ , we see that $6a^2$ will work perfectly!

STEP 4

We want to rewrite $\frac{a+5}{3a^2}$ with a denominator of $6a^2$ .
Since $6a^2 = 2 \cdot 3a^2$ , we just need to multiply the numerator and denominator by 2: $\frac{a+5}{3a^2} \cdot \frac{2}{2} = \frac{2(a+5)}{6a^2} = \frac{2a+10}{6a^2}$ Now our equation looks like this: $\frac{a-6}{6a^2} = \frac{1}{6a^2} + \frac{2a+10}{6a^2}$ Much better!

STEP 5

Now that we have a common denominator, we can combine those fractions on the right side: $\frac{a-6}{6a^2} = \frac{1 + 2a + 10}{6a^2}$ $\frac{a-6}{6a^2} = \frac{2a + 11}{6a^2}$

STEP 6

Since the denominators are the same, we can set the numerators equal to each other.
Think of it like this: if two fractions are equal and their denominators are the same, then their numerators must also be equal! $a - 6 = 2a + 11$

STEP 7

Time to isolate $a$ !
Let's subtract $a$ from both sides: $a - 6 - a = 2a + 11 - a$ $-6 = a + 11$

STEP 8

Now, subtract 11 from both sides: $-6 - 11 = a + 11 - 11$ $-17 = a$ So, $a = \mathbf{-17}$ !

STEP 9

Remember those pesky extraneous solutions?
We need to make sure our solution doesn't make the denominator zero.
If $a = -17$ , then $6a^2 = 6(-17)^2 = 6(289) = 1734$ , and $3a^2 = 3(-17)^2 = 3(289) = 867$ .
Since neither of these is zero, $a = -17$ is a valid solution!

STEP 10

$a = -17$

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QuestionSolve each equation. Remember to check for extraneous solutions.
11) $\frac{a-6}{6a^2} = \frac{1}{6a^2} + \frac{a+5}{3a^2}$

Studdy Solution

STEP 1

STEP 2

1. Find Common Denominator
2. Combine Fractions
3. Solve for a
4. Check for Extraneous Solutions

STEP 3

Alright, let's get this party started!
We've got fractions with different denominators, so our first mission is to find a common denominator.
Looking at $6a^2$ , $6a^2$ , and $3a^2$ , we see that $6a^2$ will work perfectly!

STEP 4

STEP 5

Now that we have a common denominator, we can combine those fractions on the right side: $\frac{a-6}{6a^2} = \frac{1 + 2a + 10}{6a^2}$ $\frac{a-6}{6a^2} = \frac{2a + 11}{6a^2}$

STEP 6

Since the denominators are the same, we can set the numerators equal to each other.
Think of it like this: if two fractions are equal and their denominators are the same, then their numerators must also be equal! $a - 6 = 2a + 11$

STEP 7

Time to isolate $a$ !
Let's subtract $a$ from both sides: $a - 6 - a = 2a + 11 - a$ $-6 = a + 11$

STEP 8

Now, subtract 11 from both sides: $-6 - 11 = a + 11 - 11$ $-17 = a$ So, $a = \mathbf{-17}$ !

STEP 9

Remember those pesky extraneous solutions?
We need to make sure our solution doesn't make the denominator zero.
If $a = -17$ , then $6a^2 = 6(-17)^2 = 6(289) = 1734$ , and $3a^2 = 3(-17)^2 = 3(289) = 867$ .
Since neither of these is zero, $a = -17$ is a valid solution!

STEP 10

$a = -17$

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QuestionSolve each equation. Remember to check for extraneous solutions.11) a−66a2=16a2+a+53a2\frac{a-6}{6a^2} = \frac{1}{6a^2} + \frac{a+5}{3a^2}6a2a−6​=6a21​+3a2a+5​

Studdy Solution

STEP 1

STEP 2

1. Find Common Denominator2. Combine Fractions3. Solve for *a*4. Check for Extraneous Solutions

STEP 3

Alright, let's **get this party started**!We've got fractions with different denominators, so our first mission is to find a **common denominator**.Looking at 6a26a^26a2, 6a26a^26a2, and 3a23a^23a2, we see that 6a26a^26a2 will work perfectly!

STEP 4

STEP 5

Now that we have a **common denominator**, we can **combine** those fractions on the right side: a−66a2=1+2a+106a2 \frac{a-6}{6a^2} = \frac{1 + 2a + 10}{6a^2} 6a2a−6​=6a21+2a+10​ a−66a2=2a+116a2 \frac{a-6}{6a^2} = \frac{2a + 11}{6a^2} 6a2a−6​=6a22a+11​

STEP 6

Since the denominators are the same, we can set the numerators equal to each other.Think of it like this: if two fractions are equal and their denominators are the same, then their numerators *must* also be equal! a−6=2a+11 a - 6 = 2a + 11 a−6=2a+11

STEP 7

Time to **isolate** aaa!Let's subtract aaa from both sides: a−6−a=2a+11−a a - 6 - a = 2a + 11 - a a−6−a=2a+11−a −6=a+11 -6 = a + 11 −6=a+11

STEP 8

Now, subtract 11 from both sides: −6−11=a+11−11 -6 - 11 = a + 11 - 11 −6−11=a+11−11 −17=a -17 = a −17=aSo, a=−17a = \mathbf{-17}a=−17!

STEP 9

STEP 10

a=−17a = -17a=−17

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QuestionSolve each equation. Remember to check for extraneous solutions.
11) $\frac{a-6}{6a^2} = \frac{1}{6a^2} + \frac{a+5}{3a^2}$

1. Find Common Denominator
2. Combine Fractions
3. Solve for a
4. Check for Extraneous Solutions

Alright, let's get this party started!
We've got fractions with different denominators, so our first mission is to find a common denominator.
Looking at $6a^2$ , $6a^2$ , and $3a^2$ , we see that $6a^2$ will work perfectly!

Now that we have a common denominator, we can combine those fractions on the right side: $\frac{a-6}{6a^2} = \frac{1 + 2a + 10}{6a^2}$ $\frac{a-6}{6a^2} = \frac{2a + 11}{6a^2}$

Since the denominators are the same, we can set the numerators equal to each other.
Think of it like this: if two fractions are equal and their denominators are the same, then their numerators must also be equal! $a - 6 = 2a + 11$

Time to isolate $a$ !
Let's subtract $a$ from both sides: $a - 6 - a = 2a + 11 - a$ $-6 = a + 11$

Now, subtract 11 from both sides: $-6 - 11 = a + 11 - 11$ $-17 = a$ So, $a = \mathbf{-17}$ !

$a = -17$