Question

using the linear combination method to solve a problem
lydia buys 5 pounds of apples and 3 pounds of bananas for a total of $8.50. ari buys 3 pounds of apples and 2 pounds of bananas for a total of $5.25. this system of equations represents the situation, where x is the cost per pound of apples, and y is the cost per pound of bananas.
$5x + 3y = 8.5$
$3x + 2y = 5.25$
if you multiply the first equation by 2, what number should you multiply the second equation by in order to eliminate the y terms when making a linear combination?
complete the multiplication and add the equations. what is the result?
what is the price per pound of apples? $
what is the price per pound of bananas? $

Explanation:

Step1: Eliminate y terms via scaling

To eliminate $y$, multiply the second equation by $-3$ (since first equation scaled by 2 gives $6y$, second scaled by $-3$ gives $-6y$).
First equation *2: $2(5x+3y)=2(8.5) \implies 10x + 6y = 17$
Second equation *-3: $-3(3x+2y)=-3(5.25) \implies -9x - 6y = -15.75$

Step2: Add the two new equations

Add the scaled equations to cancel $y$.
$(10x + 6y) + (-9x - 6y) = 17 + (-15.75)$
$\implies x = 1.25$

Step3: Solve for y using x value

Substitute $x=1.25$ into original second equation.
$3(1.25) + 2y = 5.25$
$\implies 3.75 + 2y = 5.25$
$\implies 2y = 5.25 - 3.75 = 1.5$
$\implies y = \frac{1.5}{2} = 0.75$

Step4: Verify combined equation result

The sum of the scaled equations is $x = 1.25$, or simplified from the addition: $x = 1.25$.

Answer:

1. Number to multiply second equation by: $-3$
2. Result of multiplication and addition: $x = 1.25$
3. Price per pound of apples: $\$1.25$
4. Price per pound of bananas: $\$0.75$