To solve for [tex]\( y \)[/tex], we start by using the information given:
1. [tex]\( LM = 4y + 3 \)[/tex]
2. [tex]\( MN = 2y + 15 \)[/tex]
3. [tex]\( LN = 42 \)[/tex]
Since point [tex]\( M \)[/tex] is between points [tex]\( L \)[/tex] and [tex]\( N \)[/tex], we know that the total distance [tex]\( LN \)[/tex] is the sum of [tex]\( LM \)[/tex] and [tex]\( MN \)[/tex]:
[tex]\[ LN = LM + MN \][/tex]
Substitute the given expressions into the equation:
[tex]\[ 42 = (4y + 3) + (2y + 15) \][/tex]
Next, combine the like terms:
[tex]\[ 42 = 4y + 3 + 2y + 15 \][/tex]
Simplify the equation:
[tex]\[ 42 = 6y + 18 \][/tex]
To isolate [tex]\( y \)[/tex], subtract 18 from both sides:
[tex]\[ 42 - 18 = 6y \][/tex]
This simplifies to:
[tex]\[ 24 = 6y \][/tex]
Finally, divide both sides by 6 to solve for [tex]\( y \)[/tex]:
[tex]\[ y = \frac{24}{6} \][/tex]
So, the value of [tex]\( y \)[/tex] is:
[tex]\[ y = 4 \][/tex]
