To solve the given equation [tex]\(\frac{x+3}{2} = \frac{3x+5}{5}\)[/tex], let's explore the method of using the multiplication property of equality to multiply both sides of the equation by 10, which effectively eliminates the denominators.
Here's the detailed, step-by-step solution:
1. Original Equation:
[tex]\[\frac{x+3}{2} = \frac{3x+5}{5}\][/tex]
2. Multiply Both Sides by 10:
To eliminate the denominators, multiply both sides of the equation by 10:
[tex]\[10 \cdot \left(\frac{x+3}{2}\right) = 10 \cdot \left(\frac{3x+5}{5}\right)\][/tex]
3. Simplifying the Multiplication:
When you multiply, it removes the denominators:
[tex]\[5 \cdot (x+3) = 2 \cdot (3x+5)\][/tex]
4. Apply the Distributive Property:
Next, distribute the constants through the parentheses:
[tex]\[5x + 15 = 6x + 10\][/tex]
5. Rearrange the Equation:
To isolate the variable [tex]\(x\)[/tex], first move all the terms involving [tex]\(x\)[/tex] to one side and constants to the other side. You can do this by subtracting [tex]\(5x\)[/tex] and [tex]\(10\)[/tex] from both sides:
[tex]\[5x + 15 - 5x - 10 = 6x + 10 - 5x - 10\][/tex]
Simplifying this gives:
[tex]\[5 = x\][/tex]
So, the solution to the equation [tex]\(\frac{x+3}{2} = \frac{3x+5}{5}\)[/tex] using the multiplication property of equality, eliminating the denominators, and solving step by step is:
[tex]\[x = 3\][/tex]
Therefore, the correct answer is using the multiplication property of equality to multiply both sides of the equation by 10.
