Question: Real Numbers
Identifying equivalent algebraic expressions
(a) Select all equivalent expressions from the list for the expression.
6(7x - 4)
42x - 4
42x - 24
13x - 4
6\cdot7x - 6\cdot4
(b) Select all equivalent expressions from the list for the expression.
10y + 2y + 9y
21y
21 + y
10y + 11y
11y - 10y
1. **State the problem:**
We need to find which expressions are equivalent to the given algebraic expressions.
2. **Problem (a):** Simplify $$6(7x - 4)$$.
3. **Use the distributive property:** $$a(b + c) = ab + ac$$.
4. Apply it:
$$6(7x - 4) = 6 \cdot 7x - 6 \cdot 4$$
5. Calculate each term:
$$6 \cdot 7x = 42x$$
$$6 \cdot 4 = 24$$
6. So,
$$6(7x - 4) = 42x - 24$$
7. **Check each option for (a):**
- $$42x - 4$$ (not equivalent, constant term differs)
- $$42x - 24$$ (equivalent)
- $$13x - 4$$ (not equivalent)
- $$6 \cdot 7x - 6 \cdot 4$$ (equivalent, same as original expression before simplification)
8. **Problem (b):** Simplify $$10y + 2y + 9y$$.
9. Combine like terms (add coefficients of $$y$$):
$$10y + 2y + 9y = (10 + 2 + 9)y = 21y$$
10. **Check each option for (b):**
- $$21y$$ (equivalent)
- $$21 + y$$ (not equivalent, addition is incorrect)
- $$10y + 11y$$ (simplifies to $$21y$$, equivalent)
- $$11y - 10y$$ (simplifies to $$y$$, not equivalent)
**Final answers:**
- (a) Equivalent expressions: $$42x - 24$$ and $$6 \cdot 7x - 6 \cdot 4$$
- (b) Equivalent expressions: $$21y$$ and $$10y + 11y$$