Students need to learn and practice three main techniques for solving systems of linear equations: graphing, addition and substitution. These three techniques -- which allow you to solve groups of equations that have more than one variable -- can be difficult for students to master. You can, however, effectively teach each of these techniques in creative ways that help your students learn, while having fun at the same time.
To teach the graphing technique, tell students that they need to lay mines along the path of an enemy navy. Explain that their battleship will be moving in a line described by one equation, such as x + y = 9, while the other navy will have ships moving in lines described by three different equations, such as x - 2y = 0, y + 1 = 5, and 2y = x + 3. Ask them to determine where they would plant the three mines so that the other battleships be certain to land on them. To do this, the students should plot each of the four paths on graph paper and find the points where the paths intersect. When they finish, discuss that each intersect is actually a solution for the two equations. For example, if the intersect is at (2, 0), then both equations involved are true when x = 2 and y = 0.
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To teach the addition technique, ask each student to pick two "secret" values for x and y and write them on a piece of paper. For example, these secret values could be x = 3 and y = 9. Then, ask the students to write two equations, one that shows the sum of x and y, and one that shows the difference of x and y. For example, the above values would produce "x + y = 12" and "x - y = -6." Tell one student to write both of her equations on the board, and then tell the other students to subtract the second from the first, which would leave, 0x + 2y = 18, or y = 9. The student will be surprised when the other students are able to guess the value of her “y.” See if any student can figure out how to find the value of her “x,” as well. Continue with other students and challenge students to create more complex equations and to solve them, using this technique.
To teach the basics of the substitution technique, write two questions on the board with the "y" in each equation on a colored sticky note. For example, you might write y = x + 3 and 3x = 2y - 1 on the board, with one sticky-note "y" in each equation. Explain that the "y" in each equation needs to have the same value for the system to be solved, which is why both sticky notes are the same color. Then, take another colored sticky note and ask students which other part of the equation is equal to y and can be written on a sticky note. In the above example, "x + 3" would be the most obvious answer. Write this on the sticky note and then discuss that anywhere that you see a y, you can replace it with x + 3, since y and x+3 are equal to each other. Then, switch the "y" sticky note from the second equation with the "x + 3" sticky note and ask the students to solve the equation.
Once the students have learned the concept of each technique, challenge the students to see which technique works most effectively for them. Divide the class into three teams, and tell each team one technique it can use to solve equations. Then, call up one student from each team to compete in a “techniques” race, in which the three students race against each other to get the correct answer to a system of equations, as quickly as possible. Repeat with the other students from each team several times, and then discuss which techniques seem to be the fastest and in which situations the other techniques might be specifically helpful.
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