Multi-step linear equations. Game on.

Distribute $4$: $4x - 8 + 3x = 13$. Combine like terms: $7x - 8 = 13$. Add $8$ to both sides: $7x = 21$. Divide by $7$: $x = 3\textsf{.}$

Distribute $\dfrac{1}{2}$: $2x + 3 + x = 15$. Combine like terms: $3x + 3 = 15$. Subtract $3$ from both sides: $3x = 12$. Divide by $3$: $x = 4\textsf{.}$

Distribute $\dfrac{1}{3}$: $2x - 3 + 2x = 11$. Combine like terms: $4x - 3 = 11$. Add $3$ to both sides: $4x = 14$. Divide by $4$: $x = \dfrac{7}{2}\textsf{.}$

Subtract $2x$ from both sides: $3x + 4 = 19$. Subtract $4$ from both sides: $3x = 15$. Divide by $3$: $x = 5\textsf{.}$

Subtract $3x$ from both sides: $4x - 3 = 17$. Add $3$ to both sides: $4x = 20$. Divide by $4$: $x = 5\textsf{.}$

Subtract $2x$ from both sides: $9 = 3x - 6$. Add $6$ to both sides: $15 = 3x$. Divide by $3$: $x = 5\textsf{.}$

Multiply every term by $2$ to clear the fraction: $x + 8 = 2x - 2$. Subtract $x$ from both sides: $8 = x - 2$. Add $2$ to both sides: $x = 10\textsf{.}$

Subtract $\dfrac{1}{3}x$ from both sides: $\dfrac{1}{3}x + 5 = 11$. Subtract $5$ from both sides: $\dfrac{1}{3}x = 6$. Multiply both sides by $3$: $x = 18\textsf{.}$