Simplify -(1/5)(-5x + 10) - 2(3 - 4x). Which of these is the result: 9x - 8, -7x - 4, 9x - 4, or -7x + 4?

Step-by-step solution

1. 1

   Treat the leading minus sign as part of the coefficient. The first factor is $-\tfrac15$, not $\tfrac15$. Reading it as $+\tfrac15$ flips the sign of both terms it multiplies and is the single most common source of the wrong options here.

2. 2

   Distribute $-\tfrac15$ across the first bracket.

   $-\frac15(-5x)=x,\qquad -\frac15(10)=-2\quad\Longrightarrow\quad x-2$

   The $-5x$ term is designed to cancel the fifth exactly, leaving a clean $x$.

3. 3

   Distribute $-2$ across the second bracket.

   $-2(3)=-6,\qquad -2(-4x)=+8x\quad\Longrightarrow\quad 8x-6$

   Note the double negative: $-2$ times $-4x$ is positive.

4. 4

   Combine like terms. Adding the two results,

   $(x-2)+(8x-6)=9x-8$

   which is option (A).

5. 5

   Test three values of $x$ against the original expression. At $x=0$: original $=-\tfrac15(10)-2(3)=-8$ and $9(0)-8=-8$. At $x=1$: original $=-\tfrac15(5)-2(-1)=-1+2=1$ and $9(1)-8=1$. At $x=3$: original $=-\tfrac15(-5)-2(-9)=1+18=19$ and $9(3)-8=19$. Three agreements settle a linear expression completely, since two points already determine a line.