Question

choose the three expressions that are equivalent to $30x + 21y$. $\square\\ 2(12x + 9y) + 6x + 3y$ $\square\\ 5(6x + 21y)$ $\square\\ 3(7y + 10x)$ $\square\\ 3(10x + 8y)$ $\square\\ 10x + 2x + 6y + 11y + 4x + 18x$ $\square\\ 17y + y + 9x + 8x + 3y + 13x$

Explanation:

Step1: Analyze \(2(12x + 9y)+6x + 3y\)

First, expand \(2(12x + 9y)\): \(2\times12x+2\times9y = 24x + 18y\). Then add \(6x + 3y\): \((24x+6x)+(18y + 3y)=30x+21y\).

Step2: Analyze \(5(6x + 21y)\)

Expand it: \(5\times6x+5\times21y = 30x+105y
eq30x + 21y\).

Step3: Analyze \(3(7y + 10x)\)

Expand it: \(3\times7y+3\times10x=21y + 30x=30x+21y\).

Step4: Analyze \(3(10x + 8y)\)

Expand it: \(3\times10x+3\times8y = 30x+24y
eq30x + 21y\).

Step5: Analyze \(10x + 2x + 6y + 11y + 4x + 18x\)

Combine like terms for \(x\): \(10x+2x + 4x+18x=(10 + 2+4 + 18)x=34x\). For \(y\): \(6y+11y = 17y\). So it is \(34x+17y
eq30x + 21y\)? Wait, no, recalculate: \(10x+2x+4x + 18x=(10 + 2+4+18)x = 34x\)? Wait, no, 10+2=12, 12+4=16, 16+18=34? Wait, no, original expression: \(10x + 2x + 6y + 11y + 4x + 18x\). Wait, 10x+2x=12x, 12x+4x=16x, 16x+18x=34x. 6y+11y=17y. Wait, that's wrong. Wait, maybe I made a mistake. Wait, no, let's check again. Wait, the original expression is \(10x + 2x + 6y + 11y + 4x + 18x\). Wait, 10x+2x=12x, 12x+4x=16x, 16x+18x=34x. 6y+11y=17y. So that's \(34x + 17y\), which is not equal. Wait, no, maybe I misread the expression. Wait, the sixth option: \(10x + 2x + 6y + 11y + 4x + 18x\). Wait, 10x+2x=12x, 12x+4x=16x, 16x+18x=34x. 6y+11y=17y. So that's not equal. Wait, but the seventh option: \(17y + y + 9x + 8x + 3y + 13x\). Combine \(x\) terms: \(9x+8x+13x=(9 + 8+13)x=30x\). Combine \(y\) terms: \(17y+y+3y=(17 + 1+3)y=21y\). So \(30x+21y\). Oh! I made a mistake in step5. Let's redo step5:

Step5 (corrected): Analyze \(10x + 2x + 6y + 11y + 4x + 18x\)

Wait, no, the seventh option is \(17y + y + 9x + 8x + 3y + 13x\). Let's do step6 for the seventh option:

Step6: Analyze \(17y + y + 9x + 8x + 3y + 13x\)

Combine \(x\) terms: \(9x+8x+13x=(9 + 8+13)x=30x\). Combine \(y\) terms: \(17y+y+3y=(17 + 1+3)y=21y\). So it is \(30x+21y\). And the sixth option: \(10x + 2x + 6y + 11y + 4x + 18x\). Let's recalculate: 10x+2x=12x, 12x+4x=16x, 16x+18x=34x. 6y+11y=17y. So that's \(34x+17y\). So my mistake earlier. So step5 was wrong, the sixth option is not, but the seventh option is. And step1: \(2(12x + 9y)+6x + 3y\) gives 30x+21y. Step3: \(3(7y + 10x)\) gives 30x+21y. Step6: \(17y + y + 9x + 8x + 3y + 13x\) gives 30x+21y. Wait, but the problem says choose three. Wait, let's re - check each:

1. \(2(12x + 9y)+6x + 3y\): 24x+18y+6x+3y=30x+21y: correct.

1. \(5(6x + 21y)\): 30x+105y: incorrect.

1. \(3(7y + 10x)\): 21y+30x=30x+21y: correct.

1. \(3(10x + 8y)\): 30x+24y: incorrect.

1. \(10x + 2x + 6y + 11y + 4x + 18x\): 34x+17y: incorrect.

1. \(17y + y + 9x + 8x + 3y + 13x\): (9x+8x+13x)+(17y + y+3y)=30x+21y: correct.

Wait, so the three correct ones are \(2(12x + 9y)+6x + 3y\), \(3(7y + 10x)\), and \(17y + y + 9x + 8x + 3y + 13x\)? Wait, but let's check the sixth option again. Wait, the sixth option is \(10x + 2x + 6y + 11y + 4x + 18x\). Let's add the x terms: 10+2+4+18=34, so 34x. y terms:6+11=17, so 17y. So that's 34x+17y. The seventh option: 17y + y+9x+8x+3y+13x. x terms:9+8+13=30, y terms:17+1+3=21. So that's 30x+21y. And the first option: 2(12x+9y)+6x+3y=24x+18y+6x+3y=30x+21y. Third option:3(7y+10x)=21y+30x=30x+21y. So the three correct expressions are:

• \(2(12x + 9y)+6x + 3y\)

• \(3(7y + 10x)\)

• \(17y + y + 9x + 8x + 3y + 13x\)

Wait, but let's check the seventh option again. 17y + y=18y, 18y+3y=21y. 9x+8x=17x, 17x+13x=30x. Yes, 30x+21y. And the first option: 212x=24x, 29y=18y, then +6x+3y: 24x+6x=30x, 18y+3y=21y. Correct. Third option: 37y=21y, 310x=30x. Correct.

Answer:

• \(2(12x + 9y)+6x + 3y\)
• \(3(7y + 10x)\)
• \(17y + y + 9x + 8x + 3y + 13x\)