To solve the equation [tex]\(\frac{x+3}{2}=\frac{3 x+5}{5}\)[/tex], a different method instead of cross multiplication could be used. Let's explore the options:
- Distributing [tex]\(x+3\)[/tex] and then [tex]\(3 x+5\)[/tex] to both sides of the equation: This method isn't straightforward in this context as we cannot directly distribute expressions across a division.
- Distributing [tex]\(x-3\)[/tex] and then [tex]\(3 x-5\)[/tex] to both sides of the equation: Similar to the previous approach, distributing [tex]\(x-3\)[/tex] and [tex]\(3 x-5\)[/tex] across a division doesn't directly help in solving the equation.
- Using the multiplication property of equality to multiply both sides of the equation by 10: Multiplying both sides of the equation by 10 is an effective and straightforward method to eliminate the denominators. Here's how:
[tex]\[
\frac{x+3}{2} = \frac{3 x+5}{5}
\][/tex]
Multiply both sides by 10:
[tex]\[
10 \cdot \frac{x+3}{2} = 10 \cdot \frac{3 x+5}{5}
\][/tex]
Simplify:
[tex]\[
5(x+3) = 2(3x+5)
\][/tex]
This then simplifies to:
[tex]\[
5x + 15 = 6x + 10
\][/tex]
Using the subtraction property of equality, subtract [tex]\(5x\)[/tex] and 10 from both sides:
[tex]\[
15 - 10 = 6x - 5x
\][/tex]
Simplify and solve for [tex]\(x\)[/tex]:
[tex]\[
x = 5
\][/tex]
- Using the multiplication property of equality to multiply both sides of the equation by [tex]\(\frac{1}{10}\)[/tex]: Multiplying by [tex]\(\frac{1}{10}\)[/tex] would complicate the equation further rather than simplifying it as we aim to clear the fractions.
Thus, the most effective alternative method to cross multiplication is:
using the multiplication property of equality to multiply both sides of the equation by 10.
