1. Let's start by stating the problem: solving simultaneous equations means finding values of variables that satisfy two or more equations at the same time. 2. Consider the system: $$\begin{cases} ax + by = c \\ dx + ey = f \end{cases}$$ where $a,b,c,d,e,f$ are constants. 3. One common method is substitution: solve one equation for one variable, then substitute into the other. 4. For example, solve the first equation for $x$: $$x = \frac{c - by}{a}$$ 5. Substitute into the second equation: $$d\left(\frac{c - by}{a}\right) + ey = f$$ 6. Multiply both sides by $a$ to clear the denominator: $$d(c - by) + aey = af$$ 7. Distribute $d$: $$dc - dby + aey = af$$ 8. Group terms with $y$: $$(-db + ae)y = af - dc$$ 9. Solve for $y$: $$y = \frac{af - dc}{ae - db}$$ 10. Substitute $y$ back into $x = \frac{c - by}{a}$ to find $x$: $$x = \frac{c - b\left(\frac{af - dc}{ae - db}\right)}{a}$$ 11. Simplify the expression for $x$ carefully. 12. This method works when $ae - db \neq 0$ (the determinant is non-zero), ensuring a unique solution. 13. Another method is elimination, where you add or subtract equations to eliminate one variable. 14. Simultaneous equations are fundamental in algebra and appear in many real-world problems. This explanation covers the basics of solving simultaneous linear equations using substitution and the key formula involved.