Question

name1-6 lesson quizlinear systems1. solve the system of equations.$\begin{cases} 4x + y = -1 \\ -5x - 2y = -4 end{cases}$2. solve the system of equations.$\begin{cases} x + y + z = 9 \\ -2x + y + 2z = 3 \\ x - 4y - z = 2 end{cases}LXB0\boldsymbol{\text{b}}$ $(3, 3, 3)LXB1\boldsymbol{\text{d}}$ $(7, 1, 1)$3. shandra is on vacation and wants to buy souvenirs for at least eight friends. a postcard book costs $2.50 and a magnet costs $4.00. she can spend up to $30 altogether. write a system of inequalities to represent the situation.4. graph the system of inequalities you wrote for item 3.5. which matrix represents the system of equations?$\begin{cases} 3x + y = -9 \\ x + 5y = 16 end{cases}$$\boldsymbol{\text{a}}$ $\begin{bmatrix} 3 & 0 \\ 0 & 5 end{bmatrix}$$\boldsymbol{\text{b}}$ $\begin{bmatrix} 3 & 1 \\ 1 & 5 end{bmatrix}$$\boldsymbol{\text{c}}$ $\begin{bmatrix} 3 & 0 &mid& -9 \\ 0 & 5 &mid& 16 end{bmatrix}$$\boldsymbol{\text{d}}$ $\begin{bmatrix} 3 & 1 &mid& -9 \\ 1 & 5 &mid& 16 end{bmatrix}$

Explanation:

Step1: Eliminate $y$ from 2-variable system

Multiply first equation by 2: $2(4x + y) = 2(-1) \implies 8x + 2y = -2$
Add to second equation: $(8x + 2y) + (-5x - 2y) = -2 + (-4)$

Step2: Solve for $x$

$3x = -6 \implies x = \frac{-6}{3} = -2$

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

Use $4x + y = -1$: $4(-2) + y = -1 \implies -8 + y = -1 \implies y = 7$

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Step1: Eliminate $z$ from equations 1&3

Add equation 1: $x + y + z = 9$ and equation 3: $x - 4y - z = 2$
$(x + y + z) + (x - 4y - z) = 9 + 2 \implies 2x - 3y = 11$ (Equation 4)

Step2: Eliminate $z$ from equations 1&2

Multiply equation1 by 2: $2x + 2y + 2z = 18$
Subtract equation2: $(2x + 2y + 2z) - (-2x + y + 2z) = 18 - 3$
$\implies 4x + y = 15$ (Equation 5)

Step3: Solve 2-variable system (4&5)

Multiply Equation5 by 3: $12x + 3y = 45$
Add to Equation4: $(2x - 3y) + (12x + 3y) = 11 + 45 \implies 14x = 56 \implies x=4$

Step4: Find $y$ with $x=4$

Use Equation5: $4(4) + y =15 \implies 16 + y=15 \implies y=-1$

Step5: Find $z$ with $x=4,y=-1$

Use Equation1: $4 + (-1) + z =9 \implies 3 + z=9 \implies z=6$

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Step1: Define variables

Let $x$ = number of postcard books, $y$ = number of magnets

Step2: Write quantity inequality

Total items ≥8: $x + y \geq 8$

Step3: Write cost inequality

Total cost ≤$30$: $2.50x + 4.00y \leq 30$

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Step1: Identify augmented matrix structure

An augmented matrix for

$$\begin{cases}a_1x + b_1y = c_1\\a_2x + b_2y = c_2\end{cases}$$

is

$$\begin{bmatrix}a_1 & b_1 &\mid& c_1\\a_2 & b_2 &\mid& c_2\end{bmatrix}$$

Step2: Match coefficients to matrix

For

$$\begin{cases}3x + y = -9\\x + 5y = 16\end{cases}$$

, substitute values into the structure.

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Answer:

1. $x=-2,\ y=7$
2. $\boldsymbol{C. (4, -1, 6)}$

3.

$$\begin{cases}x + y \geq 8\\2.5x + 4y \leq 30\end{cases}$$

(where $x\geq0,y\geq0$, non-negative integers)

1. (Graph instructions:

• Plot the line $x+y=8$ (solid, shade above the line)
• Plot the line $2.5x+4y=30$ (solid, shade below the line)
• The solution is the overlapping shaded region in the first quadrant with integer coordinates)

1. $\boldsymbol{D.

$$\begin{bmatrix}3 & 1 &\mid& -9\\1 & 5 &\mid& 16\end{bmatrix}$$

}$