Solving Systems of Equations The Beginning of Chapter 7!!!

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Presentation transcript:

Solving Systems of Equations The Beginning of Chapter 7!!!

Check out our first problem: A solution of a system of two equations in two variables is an ordered pair of real numbers that is a solution of each equation. To solve by substitution, solve one of the equations for one variable, then substitute for that variable in the other equation.

Solve the first equation for y: Second equation: Substitution step: Solve for x: Plug x into either original equation and solve for y:

More Practice: Solve the given problem by substitution, and support your answer graphically. Find the dimensions of a rectangular garden that has perimeter 100 ft and area 300 sq ft. x y by feet  How about the graph?

More Practice: Solve the given problem by substitution, and support your answer graphically. Solve the system (–3, –9), (0, 0), and (3, 9) – How about the graph?

Sometimes, solving a system algebraically involves techniques that are beyond the scope of this course. In these cases, just solve graphically! (.712, –.340) and (3.828, 1.342) Graph both functions, and calculate the intersection values… How can we be sure that these are the only two solutions?

Solving Systems by Elimination

Steps to Solving by Elimination 1. Write both equations in standard form and line up corresponding variables and constants. 2. Rewrite one or both equations so that one of the corresponding variables has opposite coefficients. 3. Add the two equations. The variable in question should be eliminated. 4. Solve for the other variable  plug answer back into either original equation to find the corresponding variable.

Look at our previous problem: 5x + 2y = 4 –3x – 2y = 0 We already have steps one and two covered!!! ( )+ 2x = 4 x = 2  y = –3

Other Examples 2x + 3y = 17 Solve the given system by elimination. –3x + y = 13 x = –2, y = 7

Other Examples x – 3y = –2 Solve the given system by elimination. 2x – 6y = 4 No Solution!!!

Other Examples 4x – 5y = 2 Solve the given system by elimination. –12x + 15y = – 6 Infinitely Many Solutions!!!

Whiteboard practice… Solve the given system by substitution.

Whiteboard Practice… Solve the given system.

Whiteboard Practice… 4x – 5y = –23 Solve the given system by elimination. 3x + 4y = 6 (x, y) = (–2, 3)

2x – y = 3 Solve the given system by elimination. –4x + 2y = 5 No Solution

Whiteboard Practice… 6x = y + 8 Solve the given system by elimination. –3y = 15x + 13 (x, y) = (1/3, – 6)

Whiteboard Practice… Solve the given system.

Whiteboard Practice… Solve the given system.

Whiteboard Practice… Solve the given system by substitution. or

Whiteboard Practice… Solve the given system by substitution.