Solve 2(5 - 4x) - 7 = -3(2x + 1) + 2(3 - x).

Step-by-step solution

1. 1

   Expand the left-hand side. Distributing the $2$ and then combining constants:

   $2(5-4x)-7=10-8x-7=3-8x.$

   Keep the $-8x$ intact: $2\times(-4x)=-8x$, and the $-7$ combines only with the $10$.

2. 2

   Expand the right-hand side. Two distributions are needed, and the first one carries a negative multiplier:

   $-3(2x+1)=-6x-3,\qquad 2(3-x)=6-2x.$

   Adding them:

   $-6x-3+6-2x=3-8x.$

   The sign trap is $-3\times1=-3$ (not $+3$) and $2\times(-x)=-2x$.

3. 3

   Compare the two sides. The equation has become

   $3-8x=3-8x.$

   Every term matches. Subtracting $3-8x$ from both sides gives $0=0$ — a statement that is true regardless of $x$, with no variable left to solve for.

4. 4

   Interpret the outcome. A linear equation ends in one of three ways. If it reduces to $x=a$ there is exactly one solution; if it reduces to a false statement such as $0=5$ there is no solution; and if it reduces to a true statement such as $0=0$ every real number is a solution. This is the third case, so the equation is an identity and its solution set is $\mathbb{R}$.

5. 5

   Confirm with sample values. At $x=0$: left $=3$, right $=-3+6=3$ ✓. At $x=1$: left $=2(1)-7=-5$, right $=-9+4=-5$ ✓. At $x=-3.5$: left $=2(19)-7=31$, right $=18+13=31$ ✓. Three arbitrary values all work, exactly as an identity predicts.