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3.2 Solving Linear Systems Algebraically

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noam
9 diapositives · 254 Views · Aug 06, 2014

Discover the fundamental methods for solving linear systems algebraically—Substitution and Linear Combination. The Substitution Method is particularly useful when one equation has a variable with a simple coefficient, allowing for easy manipulation and substitution into another equation. The Linear Combination Method involves adjusting equations to cancel variables by adding them together. Learn through practical examples, including solving pairs of equations and understanding concepts like infinite solutions and parallel lines with no solution. Master these techniques for effective problem-solving!

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3.2 Solving Linear Systems Algebraically
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  1. 3.2 Solving Linear Systems Algebraically p. 148

  2. 2 Methods for Solving Algebraically • Substitution Method (used mostly when one of the equations has a variable with a coefficient of 1 or -1) • Linear Combination Method

  3. Substitution Method • Solve one of the given equations for one of the variables. (whichever is the easiest to solve for) • Substitute the expression from step 1 into the other equation and solve for the remaining variable. • Substitute the value from step 2 into the revised equation from step 1 and solve for the 2nd variable. • Write the solution as an ordered pair (x,y).

  4. 3x-y=13 2x+2y= -10 Solve the 1st eqn for y. 3x-y=13 -y= -3x+13 y=3x-13 Now substitute 3x-13 in for the y in the 2nd equation. 2x+2(3x-13)= -10 Now, solve for x. 2x+6x-26= -10 8x=16 x=2 Now substitute the 2 in for x in for the equation from step 1. y=3(2)-13 y=6-13 y=-7 Solution: (2,-7) Plug in to check soln. Ex: Solve using substitution method

  5. Linear Combination Method • Multiply one or both equations by a real number so that when the equations are added together one variable will cancel out. • Add the 2 equations together. Solve for the remaining variable. • Substitute the value form step 2 into one of the original equations and solve for the other variable. • Write the solution as an ordered pair (x,y).

  6. 2x-6y=19 -3x+2y=10 Multiply the entire 2nd eqn. by 3 so that the y’s will cancel. 2x-6y=19 -9x+6y=30 Now add the 2 equations. -7x=49 and solve for the variable. x=-7 Substitute the -7 in for x in one of the original equations. 2(-7)-6y=19 -14-6y=19 -6y=33 y= -11/2 Now write as an ordered pair. (-7, -11/2) Plug into both equations to check. Ex: Solve using lin. combo. method.

  7. 9x-3y=15 -3x+y= -5 Which method? Substitution! Solve 2nd eqn for y. y=3x-5 9x-3(3x-5)=15 9x-9x+15=15 15=15 OK, so? What does this mean? Both equations are for the same line! ¸ many solutions Means any point on the line is a solution. Ex: Solve using either method.

  8. 6x-4y=14 -3x+2y=7 Which method? Linear combo! Multiply 2nd eqn by 2. 6x-4y=14 -6x+4y=14 Add together. 0=28 Huh? What does this mean? It means the 2 lines are parallel. No solution Since the lines do not intersect, they have no points in common. Ex: Solve using either method.

  9. Assignment