To solve the equation \(\frac{x+3}{2} = \frac{3x+5}{5}\) without using cross multiplication, we can use the multiplication property of equality to eliminate the fractions by multiplying both sides of the equation by the least common multiple (LCM) of the denominators.
In this case, the denominators are 2 and 5. The LCM of 2 and 5 is 10.
Here's the step-by-step solution using this method:
1. Start with the original equation:
[tex]\[\frac{x+3}{2} = \frac{3x+5}{5}\][/tex]
2. Multiply both sides of the equation by 10, the LCM of the denominators, to eliminate the fractions:
[tex]\[10 \cdot \left(\frac{x+3}{2}\right) = 10 \cdot \left(\frac{3x+5}{5}\right)\][/tex]
3. Simplify both sides:
[tex]\[5(x + 3) = 2(3x + 5)\][/tex]
4. Distribute 5 on the left side and 2 on the right side:
[tex]\[5x + 15 = 6x + 10\][/tex]
5. To solve for \(x\), we need to isolate the variable. First, subtract \(5x\) from both sides:
[tex]\[15 = x + 10\][/tex]
6. Next, subtract 10 from both sides:
[tex]\[5 = x\][/tex]
Thus, the solution to the equation \(\frac{x+3}{2} = \frac{3x+5}{5}\) is:
[tex]\[x = 5\][/tex]
Given the numerical result from the solution, the correct method to solve the equation without using cross multiplication is:
[tex]\[ \text{using the multiplication property of equality to multiply both sides of the equation by 10} \][/tex]
