Solve the equation -20 + 7x + 17 - 15x + 3x - 9 = 0 and give the answer as a fraction.

Step-by-step solution

1. 1

   Sort the terms into two groups. The equation mixes variable terms and constants: $7x$, $-15x$, $3x$ are the variable terms, and $-20$, $17$, $-9$ are the constants. Grouping first prevents sign slips later.

2. 2

   Add the coefficients of x. $7-15+3=-5$, so all three variable terms collapse to $-5x$. Only the coefficients add; the variable itself is unchanged.

3. 3

   Add the constants. $-20+17-9=-12$. The equation is now the two-term linear equation $-5x-12=0$.

4. 4

   Move the constant across. Adding $12$ to both sides gives $-5x=12$. Keeping the variable term where it is avoids introducing an extra sign change.

5. 5

   Divide by the coefficient -5. $x=\frac{12}{-5}=-\frac{12}{5}$. As a decimal this is $-2.4$, but the fraction $-\frac{12}{5}$ is the exact form the question asks for.

6. 6

   Substitute back to check. With $x=-\frac{12}{5}$: $7x+(-15x)+3x=-5\left(-\frac{12}{5}\right)=12$, and $12+(-12)=0$, so the original equation is satisfied exactly.