Simultaneous Equations C4B6Cb

1. **Problem Statement:** Solve the system of simultaneous equations. 2. **General Formula:** For two linear equations in two variables: $$a_1x + b_1y = c_1$$ $$a_2x + b_2y = c_2$$ We can use substitution or elimination methods. 3. **Method - Elimination:** Multiply equations to align coefficients and subtract or add to eliminate one variable. 4. **Example:** Solve: $$2x + 3y = 6$$ $$4x - y = 5$$ Multiply second equation by 3: $$3(4x - y) = 3(5)$$ $$12x - 3y = 15$$ Add to first equation: $$2x + 3y = 6$$ $$12x - 3y = 15$$ Adding: $$2x + 3y + 12x - 3y = 6 + 15$$ $$14x + \cancel{3y} - \cancel{3y} = 21$$ $$14x = 21$$ Divide both sides by 14: $$x = \frac{21}{14} = \frac{3}{2}$$ 5. **Find y:** Substitute $x=\frac{3}{2}$ into first equation: $$2\left(\frac{3}{2}\right) + 3y = 6$$ $$3 + 3y = 6$$ $$3y = 6 - 3 = 3$$ $$y = \frac{3}{3} = 1$$ 6. **Solution:** $$x = \frac{3}{2}, y = 1$$ This method can be applied to any pair of simultaneous linear equations.