Yiguo

There were an equal number of carrots, tomatoes, and cabbage in the pantry. Yiguo used 49 carrots som, some tomatoes, and some cabbages. There were 3 times as many tomatoes as cabbages left. Sixteen fewer carrots than tomatoes were left. There were 117 carrots, tomatoes, and cabbages left altogether. How many carrots were there in the pantry at first?

Correct answer:

a =  90

Step-by-step explanation:

$$\begin{gathered} a=b=c \\ y=3z \\ x=y-16 \\ x+y+z=117 \\ y=3\cdot z \\ x=y-16 \\ x+y+z=117 \\ y-3z=0 \\ x-y=-16 \\ x+y+z=117 \\ Pivot:Row1\leftrightarrow Row2 \\ x-y=-16 \\ y-3z=0 \\ x+y+z=117 \\ Row3-Row1\to Row3 \\ x-y=-16 \\ y-3z=0 \\ 2y+z=133 \\ Pivot:Row2\leftrightarrow Row3 \\ x-y=-16 \\ 2y+z=133 \\ y-3z=0 \\ Row3-\frac{1}{2}\cdot Row2\to Row3 \\ x-y=-16 \\ 2y+z=133 \\ -3.5z=-66.5 \\ z=\frac{-66.5}{-3.5}=19 \\ y=\frac{133-z}{2}=\frac{133-19}{2}=57 \\ x=-16+y=-16+57=41 \\ x=41 \\ y=57 \\ z=19 \\ a=x+49=41+49=90 \end{gathered}$$

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