One of the main concepts in algebra is the solution to a system of equations. When given two equations, it is important for students to be able to find where they intersect, as this intersection point is the solution to the system. Graphing is one way to approach this problem, and here are some activities to teach students how to find the number of solutions to a system of equations by graphing.

**1. Using Desmos:**

Desmos is an online graphing calculator which can be used to graph functions and equations. You can enter two equations and plot the functions on the same graph. This is a great tool for teaching students how to visually understand the number of solutions to a system of equations. By changing the slope and y-intercept of the second equation, you can demonstrate how the number of solutions changes.

**2. Investigation Activity:**

Give your students a set of equations and challenge them to graph and find the solutions. Then pose the question, “What if we change the constants in the equations, will the number of solutions change?” Students can experiment with changing the coefficients and y-intercepts of the equations and see what effect it has on the solutions. This is an excellent way to get students involved and thinking for themselves.

**3. Gallery Walk:**

Create a gallery of graphs with different systems of equations. Have students walk around the room and analyze the different graphs. Encourage them to make observations and record their findings. Then come together as a class to discuss the solutions of each graph. This activity is a great way to reinforce the concept of finding solutions through graphing and encourages communication and collaboration among students.

**4. Real World Problems:**

Provide students with real-world problems that can be modeled by a system of equations. For example, a bakery might want to determine how many cupcakes and muffins to sell in order to maximize profits. Students can set up the appropriate equations and then graph them to determine the solution. This type of activity makes the concept more relevant and helps students to see the value of understanding how to find solutions by graphing.

In conclusion, graphing is a powerful tool in teaching students to find the number of solutions to a system of equations. By using Desmos, investigating with different equations, hosting a gallery walk, and providing real world problems, students can be engaged and motivated to understand this concept. With practice, they will develop a deep understanding of how solutions can be found through visual representation, and be better equipped to solve algebra problems in the future.