Question

on your own

1. (some text about coins, tables, and equations)
2. liliana spends $4.00 on 2 boxes of popcorn and 2 drinks. naome spends $5.25 on 2 boxes of popcorn and 3 drinks. the prices of the popcorn and drinks can be determined by solving the system of equations shown.

\

$$\begin{cases} 2p + 2d = 4.00 \\\\ 2p + 3d = 5.25 \\end{cases}$$

a. what variable will you eliminate first? how?

b. what is the solution to the system?

c. what is the price of a box of popcorn? a drink?

1. construct arguments consider the system of equations. \

$$\begin{cases} x + y = 6 \\\\ x - y = 12 \\end{cases}$$

a. to solve by elimination, which variable would you eliminate and why?

b. will the solution be different if substitution is used instead of elimination? explain.

1. what is the solution to the system? \

$$\begin{cases} 3x + 2y = 37 \\\\ 7x - 6y = 1 \\end{cases}$$

Explanation:

Problem 4

Step1: Identify variable to eliminate

We can eliminate $p$ first, since the coefficient of $p$ is 2 in both equations. Subtract the first equation from the second equation.

$$\begin{cases} 2p + 3d = 5.25 \\ -(2p + 2d = 4.00) \end{cases}$$

Step2: Solve for $d$

Subtract the equations to cancel $p$.
$(2p-2p)+(3d-2d)=5.25-4.00$
$d=1.25$

Step3: Substitute $d$ to find $p$

Plug $d=1.25$ into $2p+2d=4.00$.
$2p + 2(1.25)=4.00$
$2p + 2.50=4.00$
$2p=4.00-2.50=1.50$
$p=\frac{1.50}{2}=0.75$

A. We can eliminate $y$ first. The coefficients of $y$ are $1$ and $-1$, so adding the two equations will cancel out $y$ directly, simplifying the calculation.
B. The solution will not be different. Both elimination and substitution are algebraic methods to solve consistent, independent systems of linear equations, so they will always yield the same unique solution.

Step1: Prepare to eliminate $y$

Multiply the first equation by 3 to make the coefficient of $y$ equal to 6, which is the opposite of $-6$ in the second equation.
$3\times(3x+2y)=3\times37$
$9x+6y=111$

Step2: Add equations to solve for $x$

Add the new equation to the second original equation to cancel $y$.

$$\begin{cases} 9x+6y=111 \\ 7x-6y=1 \end{cases}$$

$(9x+7x)+(6y-6y)=111+1$
$16x=112$
$x=\frac{112}{16}=7$

Step3: Substitute $x$ to find $y$

Plug $x=7$ into $3x+2y=37$.
$3(7)+2y=37$
$21+2y=37$
$2y=37-21=16$
$y=\frac{16}{2}=8$

Answer:

A. Eliminate $p$ first; subtract the first equation from the second (since the coefficients of $p$ are equal).
B. The solution is $(p,d)=(0.75, 1.25)$
C. Price of popcorn: $\$0.75$, Price of a drink: $\$1.25$

---

Problem 5