The problem in the image revolves around two students, Jayden and Ruby, who are saving money for a school trip by working jobs with different hourly wages. The graph and the equations provided relate their savings to the number of hours they work.

Summary:

1. Jayden's wage: Jayden starts with $24 and earns $6 per hour mowing lawns.

   • Equation: $J = 6h + 24$, where $J$ is Jayden's total savings and $h$ is the number of hours worked.

2. Ruby's wage: Ruby starts with $10 and earns $8 per hour babysitting.

   • Equation: $R = 8h + 10$, where $R$ is Ruby's total savings and $h$ is the number of hours worked.

The graph plots both of these equations, showing the total amount of money they save depending on the number of hours worked.

Questions and Answers:

a. Write the equations for Ruby and Jayden

• Already done: Jayden's equation is $J = 6h + 24$ and Ruby's equation is $R = 8h + 10$.

b. Use the graph to find where they will have saved the same amount of money

• This is where the two lines intersect. By observing the graph, you can find the point of intersection.

c. Solve these equations algebraically to confirm the solution

Let's solve the system of equations: $6h + 24 = 8h + 10$ Simplifying: $24 - 10 = 8h - 6h$ $14 = 2h$ $h = 7$ So, they will have saved the same amount of money after working for 7 hours.

To find the amount of money they saved: $J = 6(7) + 24 = 42 + 24 = 66$ So, after 7 hours, they both saved $66.

d. Check the solution graphically

From the graph, it should be confirmed that at 7 hours, both savings reach $66, which matches the algebraic solution.

e. Substitute the value $h = 7$ back into both equations to verify the savings amount is the same (already done).

Follow-up Questions:

1. What is the significance of the slope of each line in the equations?
2. How would the graph change if Jayden earned $7 per hour instead of $6?
3. What would happen if Ruby's starting savings were reduced to $5?
4. Can you identify a scenario where Ruby's line would never intersect Jayden's? What does this mean?
5. How much would each person save if they both worked 10 hours?

Tip:

When solving simultaneous equations graphically, always check the coordinates of the intersection point. This gives the exact solution for when two variables (like savings) are equal at a specific value of the third variable (hours worked).