Question

for problems 7 - 10, solve the system of equations using elimination.

1. \\(\

$$\begin{cases} 3x - 2y = 5 \\\\ x + 2y = 3 \\end{cases}$$

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1. \\(\

$$\begin{cases} 4x + 3y = 3 \\\\ 4x - 2y = 18 \\end{cases}$$

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1. \\(\

$$\begin{cases} x - y = 3 \\\\ 2x + 2y = 50 \\end{cases}$$

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1. \\(\

$$\begin{cases} 3x + 5y = 22.5 \\\\ 5x + 3y = 17.5 \\end{cases}$$

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1. stem a chemist uses the system of equations shown to find the amounts of pure water and salt water needed to make a new solution.

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$$\begin{cases} x + y = 80 \\\\ x + 0.25y = 32 \\end{cases}$$

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a. graph the system and estimate the solution.
graph with x - pure water (l), y - salt water (l), axes 0 - 50 (x), 0 - 100 (y)
b. solve the system for the precise answer. how many liters of pure water did the chemist use? how many liters of salt water?

1. open ended complete the system of equations so that it has a solution of (4, 7).

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$$\begin{cases} 5x + \\square y = 34 \\\\ 2x - \\square y = \\square \\end{cases}$$

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i’m in a learning mindset!
how was solving systems by elimination a challenge for me? is it still a challenge?

Explanation:

Problem 7

Step1: Add equations to eliminate $y$

$3x - 2y + x + 2y = 5 + 3$
$4x = 8$

Step2: Solve for $x$

$x = \frac{8}{4} = 2$

Step3: Substitute $x=2$ to find $y$

$2 + 2y = 3$
$2y = 3 - 2 = 1$
$y = \frac{1}{2} = 0.5$

Problem 8

Step1: Subtract equations to eliminate $x$

$4x + 3y - (4x - 2y) = 3 - 18$
$5y = -15$

Step2: Solve for $y$

$y = \frac{-15}{5} = -3$

Step3: Substitute $y=-3$ to find $x$

$4x + 3(-3) = 3$
$4x - 9 = 3$
$4x = 12$
$x = 3$

Problem 9

Step1: Multiply first equation by 2

$2(x - y) = 2(3)$
$2x - 2y = 6$

Step2: Add to second equation to eliminate $y$

$2x - 2y + 2x + 2y = 6 + 50$
$4x = 56$

Step3: Solve for $x$

$x = \frac{56}{4} = 14$

Step4: Substitute $x=14$ to find $y$

$14 - y = 3$
$y = 14 - 3 = 11$

Problem 10

Step1: Scale equations to eliminate $x$

Multiply first eq by 5: $15x + 25y = 112.5$
Multiply second eq by 3: $15x + 9y = 52.5$

Step2: Subtract equations to eliminate $x$

$15x + 25y - (15x + 9y) = 112.5 - 52.5$
$16y = 60$

Step3: Solve for $y$

$y = \frac{60}{16} = 3.75$

Step4: Substitute $y=3.75$ to find $x$

$3x + 5(3.75) = 22.5$
$3x + 18.75 = 22.5$
$3x = 3.75$
$x = 1.25$

Problem 11 Part B

Step1: Subtract equations to eliminate $x$

$x + y - (x + 0.25y) = 80 - 32$
$0.75y = 48$

Step2: Solve for $y$

$y = \frac{48}{0.75} = 64$

Step3: Substitute $y=64$ to find $x$

$x + 64 = 80$
$x = 80 - 64 = 16$

Problem 12

Step1: Find first missing coefficient

Substitute $x=4,y=7$: $5(4) + a(7) = 34$
$20 + 7a = 34$
$7a = 14$
$a = 2$

Step2: Choose coefficient for second equation (use 1 for simplicity)

Substitute $x=4,y=7$: $2(4) - 1(7) = 8 - 7 = 1$

Answer:

1. $x=2$, $y=0.5$
2. $x=3$, $y=-3$
3. $x=14$, $y=11$
4. $x=1.25$, $y=3.75$
5. Pure water: 16 L, Salt water: 64 L

12.

$$\begin{cases} 5x + 2y = 34 \\ 2x - 1y = 1 \end{cases}$$

(other valid coefficients for the second equation are acceptable)