Question

describe and correct any error a student may have made when solving the equation ( 0.15(y - 0.2)=2 - 0.5(1 - y) ).
( 0.15(y - 0.2)=2 - 0.5(1 - y) )
( 0.15y - 0.3=2 - 0.5+0.5y )
( 0.15y - 0.3=1.5 + 0.5y )
( 100(0.15y - 0.3)=100(1.5 + 0.5y) )
( 15y - 30=150 + 50y )
( 15y - 30 - 15y - 150=150 + 50y - 15y - 150 )
( - 180=35y )
( -\frac{180}{35}=y )

Explanation:

Step1: Distribute correctly

When distributing \(0.15(y - 0.2)\), it should be \(0.15y-0.15\times0.2 = 0.15y - 0.03\), not \(0.15y-0.3\).

Step2: Solve the corrected equation

Start with the correct equation \(0.15y-0.03=2 - 0.5(1 - y)\).
Distribute \(-0.5(1 - y)\) to get \(0.15y-0.03=2-0.5 + 0.5y\).
Simplify the right - hand side: \(0.15y-0.03 = 1.5+0.5y\).
Multiply through by \(100\) to clear decimals: \(15y-3 = 150 + 50y\).
Subtract \(15y\) from both sides: \(15y-15y - 3=150+50y-15y\), which gives \(-3=150 + 35y\).
Subtract \(150\) from both sides: \(-3-150=35y\), so \(-153 = 35y\).
Divide both sides by \(35\): \(y=-\frac{153}{35}\).

Answer:

The error was in the distribution of \(0.15(y - 0.2)\) (the student multiplied \(0.15\) by \(2\) instead of \(0.2\)). The correct solution is \(y =-\frac{153}{35}\).