Question

5.9) $\begin{cases}-3x + 6y=0\\2x + 10y=7end{cases}$
5.10) $\begin{cases}8x + 5y=-1\\-7x-3y=16end{cases}$
use the scenarios and elimination by multiplication to answer the questions.
5.11) at lunch, marc bought 2 quesadillas and 2 tacos for $10.16, and jason paid $11.84 for 5 tacos and 1 quesadilla. what was the cost of each taco and quesadilla? let the cost of a quesadilla be $x$ and the cost of a taco be $y$.
5.12) what are the coordinates of one of the vertices of a rectangle when two sides are given by the lines $x - 3y=-8$ and $3x + y=6$?

Explanation:

5.9

Step1: Multiply first - equation for elimination

Multiply the equation $-3x + 6y=0$ by $\frac{2}{3}$, we get $- 2x+4y = 0$.

Step2: Add the two equations

Add $-2x + 4y=0$ and $2x + 10y=7$.
$(-2x+4y)+(2x + 10y)=0 + 7$
$14y=7$, so $y=\frac{1}{2}$.

Step3: Substitute $y$ into the first - equation

Substitute $y = \frac{1}{2}$ into $-3x+6y = 0$.
$-3x+6\times\frac{1}{2}=0$
$-3x + 3=0$
$-3x=-3$, so $x = 1$.

Step1: Multiply equations for elimination

Multiply the equation $8x + 5y=-1$ by 3 and $-7x-3y = 16$ by 5.
The first new - equation is $24x+15y=-3$.
The second new - equation is $-35x-15y = 80$.

Step2: Add the two new equations

$(24x + 15y)+(-35x-15y)=-3 + 80$
$-11x=77$, so $x=-7$.

Step3: Substitute $x$ into the first original equation

Substitute $x=-7$ into $8x + 5y=-1$.
$8\times(-7)+5y=-1$
$-56 + 5y=-1$
$5y=55$, so $y = 11$.

Step1: Set up the system of equations

The first equation from Marc's purchase is $2x+2y=10.16$, which simplifies to $x + y=5.08$ (divide by 2).
The second equation from Jason's purchase is $x + 5y=11.84$.

Step2: Subtract the first simplified equation from the second

$(x + 5y)-(x + y)=11.84 - 5.08$
$4y=6.76$, so $y = 1.69$.

Step3: Substitute $y$ into the simplified first equation

Substitute $y = 1.69$ into $x + y=5.08$.
$x+1.69=5.08$, so $x=3.39$.

Answer:

$x = 1,y=\frac{1}{2}$

5.10