To solve the equation [tex]\(\frac{2x}{3} + 1 = \frac{7x}{15} + 3\)[/tex], we will follow a step-by-step approach:
### Step 1: Clear the fractions
To eliminate the fractions, we need to find a common denominator. The common denominator for both 3 and 15 is 15. We will now multiply every term of the equation by 15 to clear these fractions:
[tex]\[
15 \left(\frac{2x}{3}\right) + 15 \cdot 1 = 15 \left(\frac{7x}{15}\right) + 15 \cdot 3
\][/tex]
### Step 2: Simplify the equation
Now we simplify each term:
[tex]\[
5 \cdot 2x + 15 = 7x + 45
\][/tex]
This simplifies further to:
[tex]\[
10x + 15 = 7x + 45
\][/tex]
### Step 3: Get all the [tex]\(x\)[/tex] terms on one side and constants on the other
To isolate [tex]\(x\)[/tex], we subtract [tex]\(7x\)[/tex] from both sides of the equation:
[tex]\[
10x - 7x + 15 = 45
\][/tex]
This simplifies to:
[tex]\[
3x + 15 = 45
\][/tex]
We then subtract 15 from both sides:
[tex]\[
3x = 30
\][/tex]
### Step 4: Solve for [tex]\(x\)[/tex]
Finally, divide both sides by 3 to solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{30}{3}
\][/tex]
So:
[tex]\[
x = 10
\][/tex]
### Verifying the solution
Let's verify the solution by plugging [tex]\(x = 10\)[/tex] back into the original equation:
[tex]\[
\frac{2(10)}{3} + 1 = \frac{7(10)}{15} + 3
\][/tex]
Calculate each side separately:
[tex]\[
\frac{20}{3} + 1 = \frac{70}{15} + 3
\][/tex]
Simplify [tex]\(\frac{70}{15}\)[/tex] to [tex]\(\frac{14}{3}\)[/tex]:
[tex]\[
\frac{20}{3} + 1 = \frac{14}{3} + 3
\][/tex]
Convert 3 to a fraction with a denominator of 3:
[tex]\[
\frac{20}{3} + 1 = \frac{14}{3} + \frac{9}{3}
\][/tex]
This simplifies to:
[tex]\[
\frac{20}{3} + 1 = \frac{23}{3}
\][/tex]
Now add 1 on the left side (express 1 as [tex]\(\frac{3}{3}\)[/tex]):
[tex]\[
\frac{20}{3} + \frac{3}{3} = \frac{23}{3}
\][/tex]
Both sides of the equation are equal, thus confirming our solution is correct.
In conclusion, the solution to the equation [tex]\(\frac{2x}{3} + 1 = \frac{7x}{15} + 3\)[/tex] is:
[tex]\[
x = 10
\][/tex]
