Real-World Examples of Solving Systems of Equations by Graphing.

📚 Quick Study Guide

• 📈 Graphing Systems of Equations: A system of equations is two or more equations with the same variables. The solution to the system is the point where the lines intersect on a graph.
• ✏️ Steps to Solve by Graphing:
  1. Graph each equation on the same coordinate plane.
  2. Identify the point of intersection.
  3. Check the solution by substituting the coordinates into both original equations.
• 🤝 Types of Solutions:
  • One Solution: Lines intersect at one point.
  • No Solution: Lines are parallel (never intersect).
  • Infinite Solutions: Lines are the same (overlap).
• 💡 Real-World Applications: Mixture problems, break-even analysis, supply and demand.
• 📐 Slope-Intercept Form: $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Practice Quiz

1. Question 1: A coffee shop sells small coffees for $2 and large coffees for $4. On a particular morning, they sold a total of 30 coffees and made $90 in revenue. Which system of equations can be used to determine the number of small coffees (x) and large coffees (y) sold?
   1. A) $x + y = 90$, $2x + 4y = 30$
   2. B) $x + y = 30$, $2x + 4y = 90$
   3. C) $x + y = 30$, $4x + 2y = 90$
   4. D) $x - y = 30$, $2x + 4y = 90$
2. Question 2: Two friends, Alice and Bob, are saving money. Alice starts with $50 and saves $10 per week. Bob starts with $20 and saves $15 per week. After how many weeks will they have the same amount of money? This can be represented by what point of intersection on a graph?
   1. A) (5, 100)
   2. B) (6, 110)
   3. C) (7, 120)
   4. D) (8, 130)
3. Question 3: A bakery sells cookies and cakes. Cookies cost $1.50 each, and cakes cost $12 each. On Saturday, the bakery sold 50 items and made $300. How many cookies (x) and cakes (y) were sold? Which point on the graph represents this solution?
   1. A) (20, 30)
   2. B) (30, 20)
   3. C) (25, 25)
   4. D) (40, 10)
4. Question 4: A landscaping company charges $30 per hour plus a $50 service fee. Another company charges $40 per hour with no service fee. For what number of hours will the cost be the same? What does this intersection point signify?
   1. A) 3 hours
   2. B) 4 hours
   3. C) 5 hours
   4. D) 6 hours
5. Question 5: Solve the following system of equations by graphing: $y = x + 2$ and $y = -x + 4$. What is the solution (x, y)?
   1. A) (1, 3)
   2. B) (2, 4)
   3. C) (3, 1)
   4. D) (4, 2)
6. Question 6: Determine the number of solutions for the following system of equations: $y = 2x - 1$ and $y = 2x + 3$.
   1. A) One solution
   2. B) No solution
   3. C) Infinite solutions
   4. D) Two solutions
7. Question 7: A small business is deciding between two advertising strategies. Strategy A costs $200 upfront and $5 per customer. Strategy B costs $100 upfront and $10 per customer. At what number of customers will the two strategies cost the same?
   1. A) 10 customers
   2. B) 20 customers
   3. C) 30 customers
   4. D) 40 customers