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 kylas errors, if they exist.
( x + y = 3 )
( x - 3y = -1 )

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

Explanation:

Problem 14

Step1: Rewrite equations in slope-intercept form

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

Step2: Find intersection algebraically (verify graph)

Set $-p + 8 = -\frac{1}{2}p + 5$
$-p + \frac{1}{2}p = 5 - 8$
$-\frac{1}{2}p = -3$
$p = 6$
Substitute $p=6$ into $r=-p+8$:
$r = -6 + 8 = 2$

Step1: Test Kyla's solution in equations

For $x+y=3$: $-4 + (-1) = -5
eq 3$
For $x-3y=-1$: $-4 - 3(-1) = -4 + 3 = -1$ (only this works)

Step2: Find correct solution

Solve the system:
From $x+y=3$, $x=3-y$
Substitute into $x-3y=-1$:
$3 - y - 3y = -1$
$3 - 4y = -1$
$-4y = -4$
$y=1$, then $x=3-1=2$
Kyla graphed the equations incorrectly (her lines do not match the actual equations for the system).

Step1: Find equation for first line (positive slope)

The line has y-intercept $(0,3)$ and slope $m=\frac{3-0}{0-(-3)}=1$.
Equation: $y = x + 3$

Step2: Find equation for second line (negative slope)

The line has y-intercept $(0,3)$ and slope $m=\frac{3-0}{0-1.5}=-2$.
Equation: $y = -2x + 3$

Answer:

6 pounds of peanuts and 2 pounds of raisins

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Problem 15