To solve the given equation [tex]\(\frac{x+3}{2}=\frac{3x+5}{5}\)[/tex], let's consider different methods that could also solve the problem.
The original equation is:
[tex]\[
\frac{x+3}{2}=\frac{3x+5}{5}
\][/tex]
### Cross Multiplication (Method used in the student's table)
This method involves cross-multiplying to eliminate the fractions:
[tex]\[
5(x+3)=2(3x+5)
\][/tex]
### Other Methods
1. Distributing [tex]\(x+3\)[/tex] and then [tex]\(3x+5\)[/tex] to both sides of the equation:
This method does not make sense mathematically because "distributing" is not an appropriate term in this context. Distributing applies within expressions, not across an equation like this.
2. Distributing [tex]\(x-3\)[/tex] and then [tex]\(3x-5\)[/tex] to both sides of the equation:
Similar to the first point, distributing [tex]\(x-3\)[/tex] and [tex]\(3x-5\)[/tex] also does not make sense and does not follow algebraic rules correctly.
3. Using the multiplication property of equality to multiply both sides of the equation by 10:
To clear the denominators, you can multiply both sides of the equation by the least common multiple (LCM) of 2 and 5, which is 10.
[tex]\[
10 \left( \frac{x+3}{2} \right) = 10 \left( \frac{3x+5}{5} \right)
\][/tex]
Simplifying, we get:
[tex]\[
5(x+3) = 2(3x+5)
\][/tex]
This step is equivalent to what is done using cross-multiplication and will result in the same simplified equation:
[tex]\[
5x + 15 = 6x + 10
\][/tex]
Using the above steps or cross-multiplication, you would ultimately solve for [tex]\(x\)[/tex] in the same way.
4. Using the multiplication property of equality to multiply both sides of the equation by [tex]\(\frac{1}{10}\)[/tex]:
Multiplying both sides by [tex]\(\frac{1}{10}\)[/tex] will not effectively help to eliminate the denominators. Instead, it will complicate the equation further, and is not a valid approach to solving this particular type of equation.
Given the options, the correct and effective alternative method to cross-multiplication in this scenario is:
Using the multiplication property of equality to multiply both sides of the equation by 10.
