Math / Algebra

QuestionSolve the rational equation. Express numbers as integers or simplified fractions. $\frac{t+2}{3}-\frac{t+6}{5}=0$
The solution set is $\square$ \}.

Studdy Solution

STEP 1

What is this asking? We need to find the value of $t$ that makes this equation true! Watch out! Don't forget to find a common denominator when subtracting fractions.

STEP 2

1. Prepare the Equation
2. Isolate the Variable
3. Calculate the Result

STEP 3

We've got two fractions with denominators 3 and 5.
The least common denominator (LCD) is 15, which is $3 \cdot 5$ .
We'll multiply both sides of the equation by 15 to get rid of those pesky fractions.
This is like giving both sides a boost so they're on equal footing!

STEP 4

$15 \cdot \left( \frac{t+2}{3}-\frac{t+6}{5} \right) = 15 \cdot 0$ Remember, anything multiplied by zero is zero.
So the right side stays at a nice, clean zero.

STEP 5

Now, let's distribute that 15 to both terms inside the parentheses: $15 \cdot \frac{t+2}{3} - 15 \cdot \frac{t+6}{5} = 0$ We can simplify this by dividing to one: $\left(\frac{15}{3}\right) \cdot (t+2) - \left(\frac{15}{5}\right) \cdot (t+6) = 0$ $5(t+2) - 3(t+6) = 0$ See how much cleaner that looks?

STEP 6

Let's distribute the 5 and -3 to the terms in the parentheses: $5 \cdot t + 5 \cdot 2 - 3 \cdot t - 3 \cdot 6 = 0$ $5t + 10 - 3t - 18 = 0$

STEP 7

Now, gather those t terms and those constant terms: $(5t - 3t) + (10 - 18) = 0$ $2t - 8 = 0$

STEP 8

We want that t all by itself.
Let's add 8 to both sides of the equation: $2t - 8 + 8 = 0 + 8$ $2t = 8$

STEP 9

Finally, divide both sides by 2 to isolate t: $\frac{2t}{2} = \frac{8}{2}$ $t = 4$

STEP 10

We found it! $t = 4$ .
Let's check if it works in the original equation.

STEP 11

$\frac{4+2}{3}-\frac{4+6}{5} = \frac{6}{3}-\frac{10}{5} = 2 - 2 = 0$ It works!

STEP 12

The solution set is $\{4\}$ .

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QuestionSolve the rational equation. Express numbers as integers or simplified fractions. $\frac{t+2}{3}-\frac{t+6}{5}=0$
The solution set is $\square$ \}.

Studdy Solution

STEP 1

What is this asking? We need to find the value of $t$ that makes this equation true! Watch out! Don't forget to find a common denominator when subtracting fractions.

STEP 2

1. Prepare the Equation
2. Isolate the Variable
3. Calculate the Result

STEP 3

We've got two fractions with denominators 3 and 5.
The least common denominator (LCD) is 15, which is $3 \cdot 5$ .
We'll multiply both sides of the equation by 15 to get rid of those pesky fractions.
This is like giving both sides a boost so they're on equal footing!

STEP 4

$15 \cdot \left( \frac{t+2}{3}-\frac{t+6}{5} \right) = 15 \cdot 0$ Remember, anything multiplied by zero is zero.
So the right side stays at a nice, clean zero.

STEP 5

STEP 6

Let's distribute the 5 and -3 to the terms in the parentheses: $5 \cdot t + 5 \cdot 2 - 3 \cdot t - 3 \cdot 6 = 0$ $5t + 10 - 3t - 18 = 0$

STEP 7

Now, gather those t terms and those constant terms: $(5t - 3t) + (10 - 18) = 0$ $2t - 8 = 0$

STEP 8

We want that t all by itself.
Let's add 8 to both sides of the equation: $2t - 8 + 8 = 0 + 8$ $2t = 8$

STEP 9

Finally, divide both sides by 2 to isolate t: $\frac{2t}{2} = \frac{8}{2}$ $t = 4$

STEP 10

We found it! $t = 4$ .
Let's check if it works in the original equation.

STEP 11

$\frac{4+2}{3}-\frac{4+6}{5} = \frac{6}{3}-\frac{10}{5} = 2 - 2 = 0$ It works!

STEP 12

The solution set is $\{4\}$ .

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QuestionSolve the rational equation. Express numbers as integers or simplified fractions. t+23−t+65=0\frac{t+2}{3}-\frac{t+6}{5}=03t+2​−5t+6​=0The solution set is □\square□ \}.

Studdy Solution

STEP 1

What is this asking? We need to find the value of ttt that makes this equation true! Watch out! Don't forget to find a common denominator when subtracting fractions.

STEP 2

1. Prepare the Equation2. Isolate the Variable3. Calculate the Result

STEP 3

We've got two fractions with denominators **3** and **5**.The least common denominator (LCD) is **15**, which is 3⋅53 \cdot 53⋅5.We'll multiply *both* sides of the equation by **15** to get rid of those pesky fractions.This is like giving both sides a boost so they're on equal footing!

STEP 4

15⋅(t+23−t+65)=15⋅015 \cdot \left( \frac{t+2}{3}-\frac{t+6}{5} \right) = 15 \cdot 015⋅(3t+2​−5t+6​)=15⋅0 Remember, anything multiplied by zero is zero.So the right side stays at a nice, clean **zero**.

STEP 5

STEP 6

Let's distribute the **5** and **-3** to the terms in the parentheses: 5⋅t+5⋅2−3⋅t−3⋅6=05 \cdot t + 5 \cdot 2 - 3 \cdot t - 3 \cdot 6 = 05⋅t+5⋅2−3⋅t−3⋅6=0 5t+10−3t−18=05t + 10 - 3t - 18 = 05t+10−3t−18=0

STEP 7

Now, gather those *t* terms and those constant terms: (5t−3t)+(10−18)=0(5t - 3t) + (10 - 18) = 0(5t−3t)+(10−18)=0 2t−8=02t - 8 = 02t−8=0

STEP 8

We want that *t* all by itself.Let's add **8** to both sides of the equation: 2t−8+8=0+82t - 8 + 8 = 0 + 82t−8+8=0+8 2t=82t = 82t=8

STEP 9

Finally, divide both sides by **2** to isolate *t*: 2t2=82\frac{2t}{2} = \frac{8}{2}22t​=28​ t=4t = 4t=4

STEP 10

We found it! t=4t = 4t=4.Let's check if it works in the original equation.

STEP 11

4+23−4+65=63−105=2−2=0\frac{4+2}{3}-\frac{4+6}{5} = \frac{6}{3}-\frac{10}{5} = 2 - 2 = 034+2​−54+6​=36​−510​=2−2=0 It works!

STEP 12

The solution set is {4}\{4\}{4}.

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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.

QuestionSolve the rational equation. Express numbers as integers or simplified fractions. $\frac{t+2}{3}-\frac{t+6}{5}=0$
The solution set is $\square$ \}.

What is this asking? We need to find the value of $t$ that makes this equation true! Watch out! Don't forget to find a common denominator when subtracting fractions.

1. Prepare the Equation
2. Isolate the Variable
3. Calculate the Result

We've got two fractions with denominators 3 and 5.
The least common denominator (LCD) is 15, which is $3 \cdot 5$ .
We'll multiply both sides of the equation by 15 to get rid of those pesky fractions.
This is like giving both sides a boost so they're on equal footing!

$15 \cdot \left( \frac{t+2}{3}-\frac{t+6}{5} \right) = 15 \cdot 0$ Remember, anything multiplied by zero is zero.
So the right side stays at a nice, clean zero.

Let's distribute the 5 and -3 to the terms in the parentheses: $5 \cdot t + 5 \cdot 2 - 3 \cdot t - 3 \cdot 6 = 0$ $5t + 10 - 3t - 18 = 0$

Now, gather those t terms and those constant terms: $(5t - 3t) + (10 - 18) = 0$ $2t - 8 = 0$

We want that t all by itself.
Let's add 8 to both sides of the equation: $2t - 8 + 8 = 0 + 8$ $2t = 8$

Finally, divide both sides by 2 to isolate t: $\frac{2t}{2} = \frac{8}{2}$ $t = 4$

We found it! $t = 4$ .
Let's check if it works in the original equation.

$\frac{4+2}{3}-\frac{4+6}{5} = \frac{6}{3}-\frac{10}{5} = 2 - 2 = 0$ It works!

The solution set is $\{4\}$ .