Question

1. sandeep ordered peanuts and raisins for his bakery. he ordered a total of eight pounds of these ingredients. peanuts cost $1 per pound, and raisins cost $2 per pound. he spent a total of $10. the system ( p + r = 8 ) and ( p + 2r = 10 ) represents this situation. graph the system to determine how many pounds of peanuts and raisins sandeep ordered.
2. critique the reasoning of others.

kyla was asked to solve the system of equations below. she made the graph shown and stated that the solution of the system is ((-4, -1)). is kyla correct? justify your response and identify kyla’s errors, if they exist.
( x + y = 3 )
( x - 3y = -1 )

1. write the system of equations represented by this graph.

Explanation:

Question 14

Step1: Rewrite equations to slope-intercept

For $p + r = 8$:
$r = -p + 8$
For $p + 2r = 10$:
$2r = -p + 10 \implies r = -\frac{1}{2}p + 5$

Step2: Find intercepts for graphing

For $r = -p + 8$:

• When $p=0$, $r=8$; when $r=0$, $p=8$

For $r = -\frac{1}{2}p + 5$:

• When $p=0$, $r=5$; when $r=0$, $p=10$

Step3: Find intersection (solve system)

Substitute $r = -p + 8$ into $p + 2r = 10$:
$p + 2(-p + 8) = 10$
$p - 2p + 16 = 10$
$-p = 10 - 16 = -6 \implies p=6$
Then $r = -6 + 8 = 2$

First, solve the system algebraically to verify. Substitute $x = 3 - y$ (from $x+y=3$) into $x-3y=-1$:
$(3 - y) - 3y = -1 \implies 3 - 4y = -1 \implies -4y = -4 \implies y=1$. Then $x=3-1=2$, so the correct solution is $(2,1)$.
Kyla incorrectly graphed the equations: she plotted lines that intersect at $(-4,-1)$ instead of the lines corresponding to the given system, leading to a wrong solution.

Step1: Find equation for first line

Line with positive slope: passes through $(-5,-3)$ and $(0,2)$
Slope $m_1 = \frac{2 - (-3)}{0 - (-5)} = \frac{5}{5} = 1$
Y-intercept $b=2$, so equation: $y = x + 2$

Step2: Find equation for second line

Line with negative slope: passes through $(0,4)$ and $(2,0)$
Slope $m_2 = \frac{0 - 4}{2 - 0} = \frac{-4}{2} = -2$
Y-intercept $b=4$, so equation: $y = -2x + 4$

Answer:

Sandeep ordered 6 pounds of peanuts and 2 pounds of raisins.

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Question 15