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Systems of Equations Systems of Equations • Finding all values of all variables that make the equations true. What values make an equation true? 7x-3y=12 What values make an equation true? y=4x+1 What values make all equations true? y=4x+1 7x-3y=12 What values make all equations true? y=4x+1 7x-3y=12 (x,y)=(-3,-11) e) (1,1) “y=4x+1” is pronounced “the number y IS THE SAME NUMBER AS 4x+1” Solving with substitution • • • • • • • • 7x-3y=12 y=4x+1 7x-3(4x+1)=12 7x-12x-3=12 -5x-3=12 +3=+3 -5x=15 x=-3 Original Problem Original Problem Anywhere I see a y, I can write (4x+1) Distributive property (un)distributing add the same thing to both sides Divide both sides by -5 But wait, I’m not done! • • • • • • 7x-3y=12 y=4x+1 x=-3 y=4(-3)+1 y=-11 (x,y)=(-3,-11) Original Problem Original Problem Solution so far. Anywhere I see an “x”, I can write “(-3)” “2x+3y=3” is pronounced “2x+3y IS THE SAME NUMBER AS 3” b Infinitely many solutions • If you wind up with something that is ALWAYS true, you have infinitely many solutions • 2x+y=1 GIVEN • 4x+2y=2 GIVEN • (-2)(2x+y)=(-2)(1)Multiply both sides by -2 • -4x-2y=-2 New Equation • 4x+2y=2 Add to original • -------------• 0x+0y=0 • 0=0 Always true infinitely many solutions No solutions • If you wind up with something that is NEVER true, you have infinitely many solutions • 2x+y=3 GIVEN • 4x+2y=2 GIVEN • (-2)(2x+y)=(-2)(3) Multiply both sides by -2 • -4x-2y=-6 New Equation • 4x+2y=2 Add to original • -------------• 0x+0y=-4 • 0=-4 Never true No solutions Find the solution: 2x-3y = 1 6y-4x = -2 A) B) C) D) x = 2, y = 1 x =0, y = -1/3 x = 6, y = 5/3 Both A & B and infinitely many other solutions E) Both B & C and infinitely many other solutions Solution • • • • • • • • • • • 2x-3y = 1 6y-4x = -2 (2)(2x-3y) =(2)(1) multiply both sides 4x-6y =2 6y-4x = -2 add to original -------------------4x-4x+6y-6y=2-2 0=0, infinitely many solutions But is the answer D (A&B) or E (B&C)? Plug in A (2,1) and check: 2(2)-3(1)=1 A is a solution The answer is D