Equivalent Expressions 0Cbf04

1. **Problem statement:** (a) Find expressions equivalent to $$5x + 25$$ from the list: $$5(x + 5), 5\cdot x + 5\cdot 5, 5(5x + 1), 30x$$. (b) Find expressions equivalent to $$10 + 13y - 7 - y$$ from the list: $$3y + 12y, 12 + 3y, 3y + 12, 12y + 3$$. 2. **Recall the distributive property and combining like terms:** - Distributive property: $$a(b + c) = ab + ac$$. - Combine like terms by adding or subtracting coefficients of the same variable. 3. **Part (a) simplification and checking equivalence:** - Original expression: $$5x + 25$$. - Check $$5(x + 5)$$: $$5(x + 5) = 5 \cdot x + 5 \cdot 5 = 5x + 25$$, which matches the original. - Check $$5 \cdot x + 5 \cdot 5$$: This is exactly $$5x + 25$$, so equivalent. - Check $$5(5x + 1)$$: $$5(5x + 1) = 25x + 5$$, which is not equal to $$5x + 25$$. - Check $$30x$$: This is just $$30x$$, which is not equal to $$5x + 25$$. 4. **Part (b) simplification and checking equivalence:** - Original expression: $$10 + 13y - 7 - y$$. - Combine like terms: $$10 - 7 = 3$$ $$13y - y = 12y$$ So, simplified expression is $$3 + 12y$$. - Check $$3y + 12y$$: $$3y + 12y = 15y$$, not equal to $$3 + 12y$$. - Check $$12 + 3y$$: This is $$12 + 3y$$, not equal to $$3 + 12y$$. - Check $$3y + 12$$: This is $$3y + 12$$, which is the same as $$12 + 3y$$, not equal to $$3 + 12y$$. - Check $$12y + 3$$: This is $$12y + 3$$, which matches $$3 + 12y$$ (order does not matter in addition), so equivalent. 5. **Final answers:** - (a) Equivalent expressions: $$5(x + 5), 5 \cdot x + 5 \cdot 5$$. - (b) Equivalent expression: $$12y + 3$$.