1. **State the problem:** Find the lowest common multiple (LCM) of 6 and 8. 2. **Formula and rules:** The LCM of two numbers is the smallest positive integer divisible by both numbers. To find it, we can use the prime factorization method or the formula: $$\text{LCM}(a,b) = \frac{|a \times b|}{\text{GCD}(a,b)}$$ where GCD is the greatest common divisor. 3. **Calculate GCD of 6 and 8:** - Prime factors of 6: $2, 3$ - Prime factors of 8: $2, 2, 2$ - Common prime factors: $2$ - So, $\text{GCD}(6,8) = 2$ 4. **Calculate LCM:** $$\text{LCM}(6,8) = \frac{6 \times 8}{2} = \frac{48}{2} = 24$$ 5. **State the problem:** Solve the simultaneous equations: $$6x + 7y = 27$$ $$8x + 11y = 41$$ by first multiplying each equation by a different constant so that the coefficient of $x$ in both equations is 24 (the LCM from part a). 6. **Multiply equations:** - Multiply the first equation by 4: $$4 \times (6x + 7y) = 4 \times 27 \Rightarrow 24x + 28y = 108$$ - Multiply the second equation by 3: $$3 \times (8x + 11y) = 3 \times 41 \Rightarrow 24x + 33y = 123$$ 7. **Subtract the first new equation from the second to eliminate $x$:** $$ (24x + 33y) - (24x + 28y) = 123 - 108 $$ $$ 5y = 15 $$ $$ y = 3 $$ 8. **Substitute $y=3$ into the first original equation:** $$ 6x + 7(3) = 27 $$ $$ 6x + 21 = 27 $$ $$ 6x = 6 $$ $$ x = 1 $$ **Final answer:** $$\text{LCM}(6,8) = 24$$ $$x = 1, y = 3$$