To solve the proportion [tex]\(\frac{2x + 1}{3} = \frac{x + 5}{2}\)[/tex], follow these detailed steps:
1. Cross-Multiply: To eliminate the fractions, cross-multiply the terms of the equation. This means:
[tex]\[
2 \times (2x + 1) = 3 \times (x + 5)
\][/tex]
2. Distribute the Numbers: Expand both sides of the equation by distributing the coefficients:
[tex]\[
2 \times (2x + 1) \Rightarrow 2 \times 2x + 2 \times 1 = 4x + 2
\][/tex]
[tex]\[
3 \times (x + 5) \Rightarrow 3 \times x + 3 \times 5 = 3x + 15
\][/tex]
Now the equation looks like:
[tex]\[
4x + 2 = 3x + 15
\][/tex]
3. Isolate the Variable [tex]\( x \)[/tex]: To solve for [tex]\( x \)[/tex], move all the [tex]\( x \)[/tex]-terms to one side and constant terms to the opposite side. First, subtract [tex]\( 3x \)[/tex] from both sides:
[tex]\[
4x + 2 - 3x = 3x + 15 - 3x
\][/tex]
Simplifying this, we get:
[tex]\[
x + 2 = 15
\][/tex]
4. Solve for [tex]\( x \)[/tex]: Next, isolate [tex]\( x \)[/tex] by subtracting 2 from both sides:
[tex]\[
x + 2 - 2 = 15 - 2
\][/tex]
[tex]\[
x = 13
\][/tex]
Therefore, the value of [tex]\( x \)[/tex] is:
[tex]\[
x = 13
\][/tex]
