1. **State the problem:** Solve the system of linear equations for the variables. 2. **General approach:** For a system of linear equations, we can use substitution, elimination, or matrix methods to find the values of the variables. 3. **Example system:** Suppose the system is: $$\begin{cases} a_1x + b_1y = c_1 \\ a_2x + b_2y = c_2 \end{cases}$$ 4. **Elimination method:** Multiply equations if necessary to align coefficients, then add or subtract to eliminate one variable. 5. **Substitution method:** Solve one equation for one variable, then substitute into the other. 6. **Show intermediate work:** For example, if we solve the first equation for $x$: $$x = \frac{c_1 - b_1y}{a_1}$$ Substitute into the second: $$a_2 \left( \frac{c_1 - b_1y}{a_1} \right) + b_2y = c_2$$ 7. **Simplify and solve for $y$:** $$\frac{a_2 c_1}{a_1} - \frac{a_2 b_1}{a_1} y + b_2 y = c_2$$ Group $y$ terms: $$\left(- \frac{a_2 b_1}{a_1} + b_2 \right) y = c_2 - \frac{a_2 c_1}{a_1}$$ 8. **Cancel common factors if any:** $$y = \frac{c_2 - \frac{a_2 c_1}{a_1}}{- \frac{a_2 b_1}{a_1} + b_2} = \frac{c_2 - \frac{a_2 c_1}{a_1}}{b_2 - \frac{a_2 b_1}{a_1}}$$ 9. **Simplify complex fractions:** $$y = \frac{c_2 a_1 - a_2 c_1}{b_2 a_1 - a_2 b_1}$$ 10. **Find $x$ by substituting $y$ back:** $$x = \frac{c_1 - b_1 y}{a_1}$$ 11. **Final answer:** $$x = \frac{c_1 (b_2 a_1 - a_2 b_1) - b_1 (c_2 a_1 - a_2 c_1)}{a_1 (b_2 a_1 - a_2 b_1)}$$ $$y = \frac{c_2 a_1 - a_2 c_1}{b_2 a_1 - a_2 b_1}$$ This is the general solution for the system of two linear equations in two variables. If you provide the specific system, I can solve it step-by-step.