Helping my daughter with her homework: solving an algebra word problem. [closed]

$x$: weight of a bag of apples (in pounds)

$y$: weight of a bag of oranges (in pounds)

First we "translate" the givens into algebraic equations:

• $(1)$ "Three bags of apples and two bags of oranges weigh $32$ pounds." $\implies 3x + 2y = 32$.
• $(2)$ "Four bags of apples and three bags of oranges weigh $44$ pounds." $\implies 4x + 3y = 44$

This gives us the system of two equations in two unknowns: $$3x + 2y = 32\tag{1}$$ $$4x + 3y = 44\tag{2}$$

Ask your daughter to solve the system of two equations in two unknowns to determine the values of $x$ and $y$.

Hints for your daughter:

• multiply equation $(1)$ by $3$, and multiply equation $(2)$ by $2$:

$$9x + 6y = 96\tag{1.1}$$ $$8x + 6y = 88\tag{2.1}$$

• subtract equation $(2.1)$ from equation $(1.1)$, which will give the value of $x$.

• Solve for $y$ using either equation $(1)$ or $(2)$ and your value for $x$.

• Then determine what $2x + y$ equals. That will be your (her) solution.

Let one bag of apple weight $x$ pounds and one bag of orange weight $y$ pounds. We then have \begin{align} 3x+2y & = 32\\ 4x+3y & = 44 \end{align} We need the weight of $2$ bags of apples and $1$ bag of orange i.e. we need $2x+y$. Note that $$2x+y = 2(3x+2y) - (4x+3y) = 2 \times 32 - 44 = 20$$