The provided image describes a problem related to solving systems of linear equations, specifically to determine the cost of items (chips and juice) based on purchases made by Apu.

Breakdown of the Problem:

1. Equations from Text:

   • From the first week's order:
     $4x + 8y = 46$,
     where $x$ is the cost of one box of chips, and $y$ is the cost of one gallon of juice.
   • From the second week's order:
     $12x + 24y = 108$,
     where Apu triples his order.

2. Graph Information:

   • The two equations are plotted on the graph:
     $4x + 8y = 46$ (or $x + 2y = 11.5$ simplified),
     $12x + 24y = 108$ (or $x + 2y = 9$ simplified).

Questions:

1. How many solutions does this graph have?

   • Simplifying the two equations:
     • First equation: $x + 2y = 11.5$
     • Second equation: $x + 2y = 9$
   • These two lines are parallel since they have the same slope ($-\frac{1}{2}$) but different y-intercepts.
     Therefore, there are no solutions as the lines never intersect.

2. Is there enough information to find the cost of chips and juice?

   • No, there is not enough information because the equations represent parallel lines, meaning they are inconsistent and do not have a common solution.

Would you like further details or a step-by-step graph analysis? Here are some follow-up questions and a tip:

Related Questions:

1. How do we determine if two equations represent parallel lines algebraically?
2. What happens graphically when two lines have the same slope but different y-intercepts?
3. How could additional information (e.g., a new equation) help solve this problem?
4. What are the general steps to check if a system of linear equations has one solution, no solution, or infinitely many solutions?
5. How can graphing calculators or tools assist in solving and visualizing systems of equations?

Tip:

Always simplify equations to their standard or slope-intercept form to easily compare slopes and y-intercepts when analyzing solutions to a system of equations.