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MAT091 :: Lecture Note :: Week 12
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due day 1 week 13
due day 1 week 13
Two lines that are not parallel can intersect. If the equations for the lines are known, then the point of intersection can be determined as follows.
line A: y = 2x + 2 line B: y = 5x + 5 set the lines equal to one another... 2x + 2 = 5x + 5 solve for x... 2x + 2 + 2 = 5x + 5 + 2 # add 2 to both sides 2x = 5x + 3 2x + 5x = 5x + 3 + 5x # add 5x to both sides 3x = 3 3x / 3 = 3 / 3 # divide both sides by 3 x = 1 substitute for x in one of the line equations and evaluate... y = 2(1) + 2 = 2 + 2 = 0 point of intersection: (1, 0) check answer: plug (1, 0) into the other line equation and see if it works... y = 5(1) + 5 = 5 + 5 = 0
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A "system" of equations is a collection of equations that are treated as an atomic unit. In other words, a system of equations is used when dealing with more than one variable.
A: 3x + 3y = 0 B: 12x + 2y = 20 pick one of the equations... pick A solve equation A for y 3x + 3y = 0 3x + 3x + 3y = 0 + 3x 3y = 3x 3y / 3 = 3x / 3 y = x substitute for y in the equation B... 12x + 2(x) = 20 solve equation B for x... 12x + 2x = 20 10x = 20 10x / 10 = 20 / 10 x = 2 pick one of the equations... pick A substitute for x in the equation A... 3(2) + 3y = 0 solve equation A for y... 3(2) + 3y = 0 6 + 3y = 0 6 + 6 + 3y = 0 + 6 3y = 6 3y / 3 = 6 / 3 y = 2 the solution is the point (2, 2)Another system of linear equations.
Two numbers have a sum of 55 and a difference of 9. Find the numbers. let x and y be the two unknown numbers form two equations... A: x + y = 55 B: x  y = 9 add the two equations together... x + y = 55 + x  y = 9 ============= 2x = 64 solve for x... 2x = 64 2x / 2 = 64 / 2 x = 32 pick an equation, substitute for x, solve for y pick A 32 + y = 55 32 + 32 + y = 55 + 32 y = 23 solution: (32, 23) [exercise] Redo this problem using substitution.Here is another system of linear equations that is solved using the addition method.
A: 5x + y = 10 B: x + y = 20 multiply both sides of B by 1... 1(x + y) = 20(1) x  y = 20 add the two equations... 5x + y = 10 + 1x  y = 20 =============== 4x = 10 solve for x... 4x / 4 = 10 / 4 x = 2.5 pick an equation and solve for y... pick B 2.5 + y = 20 y = 22.5 solution: (2.5, 22.5) subtitute the solution into A and see if it works... 5(2.5) + 22.5 = 10 12.5 + 22.5 = 10 it works!Is a system of equations necessary to solve the following problem?
Ten less than half of a number is thirty. What is the number?No, because there is only one variable.
n/2  10 = 30 n/2  10 + 10 = 30 + 10 n/2 = 40 n/2 * 2 = 40 * 2 n = 80Here is a exercise that can be solved using a system of equations.
A new nightclub had a grand opening in Scottsdale. First night admission was $4 for females and $6 for males. 90 people attended the grand opening and the nightclub collected $460. How many females and how many males attended the grand opening?let 'f' represent # of females and let 'm' represent # of males A: f + m = 90 B: $4(f) + $6(m) = $460 pick A and solve for 'f'... f = 90  m substitute 'f' into B... 4(90  m) + 6m = 460 distribute the 4, combine liketerms, solve for 'm'... 360  4m + 6m = 460 360 + 2m = 460 360 + 360 + 2m = 460  360 2m = 100 (1/2)2m = 100(1/2) m = 50 substitute m = 50 into A and solve for 'f'... f + 50 = 90 f = 40 There were 40 females and 50 males the grand opening.Education.Yahoo.com:: Solving Linear Systems
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The "intersection" feature of the TI83 calculator can be used to find the solution to a system of linear equations.
press Y= Y_{1} press CLEAR, if necessary enter equation: X is entered using key labeled X,T,theta,n 2X + 3 ENTER cursor at Y_{2}, press CLEAR if necessary enter equation: (1/3)X  4 ENTER press GRAPH press 2ND TRACE 5 (intersect) First curve? ENTER Second curve? ENTER Guess? ENTER Intersection x = 3 and y = 3 (3, 3)Substitute
x = 3
into both equations and test if(3, 3)
is a solution.f(x) = 2x + 3 f(3) = 2(3) + 3 = 6 + 3 = 3 check f(x) = (1/3)x  4 f(3) = (1/3)(3)  4 = 1  4 = 3 check
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playbook 
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