To solve the equation [tex]\(\frac{x+3}{2} = \frac{3x+5}{5}\)[/tex], a method other than cross multiplication can be used to eliminate the fractions and simplify solving for the variable. Let's examine one such method: using the multiplication property of equality.
Step-by-Step Solution:
1. Original Equation:
[tex]\[
\frac{x+3}{2} = \frac{3x+5}{5}
\][/tex]
2. Multiplication Property of Equality:
To eliminate the fractions, we can multiply both sides of the equation by the least common multiple (LCM) of the denominators. The denominators here are 2 and 5. The LCM of 2 and 5 is 10. Therefore, we multiply both sides of the equation by 10:
[tex]\[
10 \cdot \frac{x+3}{2} = 10 \cdot \frac{3x+5}{5}
\][/tex]
3. Simplifying:
Now, distribute the multiplication on both sides:
[tex]\[
\left(10 \cdot \frac{1}{2}\right) (x+3) = \left(10 \cdot \frac{1}{5}\right) (3x+5)
\][/tex]
Simplifying the coefficients:
[tex]\[
5(x+3) = 2(3x+5)
\][/tex]
4. Distributive Property:
Apply the distributive property on both sides:
[tex]\[
5x + 15 = 6x + 10
\][/tex]
5. Solving for [tex]\(x\)[/tex]:
Now, let's simplify the equation to isolate [tex]\(x\)[/tex]. First, subtract [tex]\(5x\)[/tex] from both sides:
[tex]\[
15 = x + 10
\][/tex]
Then, subtract 10 from both sides:
[tex]\[
5 = x
\][/tex]
So, the solution is:
[tex]\[
x = 5
\][/tex]
In conclusion, the method of using the multiplication property of equality to multiply both sides of the equation by 10 effectively eliminates the denominators and results in the same solution as cross multiplication.
