Algebra 2 Simultaneous Equations Line Equations Line graphs

Algebra (2) Simultaneous Equations

Line Equations • Line graphs can be described using equations This graph follows the equation y=x Y is the y axis going up X is the x axis going across So if y=1 (up) then x has to be 1 (across) If x is -1, what will y be? -1

Questions • • • Which line would be steeper? Y=2 x +1 Y=3 x Y=½ x Answer= Y= 3 x

Drawing Graphsy • Draw a graph for Y= x + 1 • Make a table for the points x y 0 1 2 3 4 5 4 3 2 1 -4 -3 -2 -1 -2 -3 -4 1 2 3 4 x

Gradients • This is how steep the graph line is • In an equation the number before the x tells you the gradient • E. g. • 3 x +1 and 3 x – 4 are parallel • The numbers afterwards tell you where the lines cross the y axis • +1 for the first line and -4 for the second

Questions • • Are these parallel 3 x -2 and 2 x -3 3 x + 1 and 4 x + 1 2 x -3 and 2 x + 4 What is the equation for the red line? Y = 2 x + 4

Answers • No • Yes • Y = 2 x -2

Do these points lie on this line? • Does the point (2, 5), lie on the line y=x+4 • You do not need to draw the graph, just workout the equation • Y=2+4 • Y=6 • This would be (2, 6) not (2, 5)

Questions • • • Do these points lie on these lines? (3, 7) on y = x + 4 (2, 6) on y = 2 x + 3 (1, 6) on y = 3 x -1 (0, -0. 5) on y = 2 x – 5 (4, 16) on y = 5 x - 3

Answers • Yes • No

Drawing Graphs the Quick Way • • Draw a graph for 2 x + y = 6 Find out where x= 0 and y = 0 are First x = 0 0 + 6 = 6 so (0, 6) Now y = 0 2 x + 0 = 6 X= 3 (3, 0) 2 x + y = 6

Points of Intersection • Two lines cross at a point of intersection • Alex and Tom are buying some food at a youth club • Alex buys two biscuits and 1 drink for 10 p • Tom buys one biscuit and two drinks for 14 p • Alex’s equation is 2 b + d = 10 • Tom’s is b + 2 d= 14 • How much is a biscuit and how much is a drink?

Example d Alex: 2 b + d =10 When b= 0 d =10 6 p (0, 10) When d= 0, b = 5 (5, 0) Tom: b + 2 d = 14 When b = 0, d = 7 (0, 7) When d = 0, 14 (14, 0) 2 p b The cost of 1 drink is 6 p The cost of 1 biscuit is 2 p

Simultaneous Equations • When you solve two equations at the same time you have done a simultaneous equation • Example • Draw graphs to solve these equations • X+y=5 • 2 x + 4 y = 12

Example X+y=5 If x= 0 then y = 5 Answer = (4, 1) (0, 5) If y = 0, then x = 5 (5, 0) 2 x + 4 y = 12 If x= 0, then y = 3 (0, 3) If y = 0, then x = 6 (6, 0) 2 x + 4 y = 12 X+y=5

Solving Simultaneous Equations • • • • Solve these simultaneous equations 5 x + y = 13 X+y=5 Get rid of the y by subtracting the two equations 5 x + y = 13 X+y=5 4 x = 8 X=2 Now put x=2 into one of your equations the find out what y is 5 x + y = 13 10 + y = 13 Y=3 So x = 2, y = 3

Solving Simultaneous Equations • • • • When subtracting does not work you have to add 2 x + y = 8 X–y=7 You have to add to get rid of the y 2 x + y = 8 X–y=7 3 x = 15 X=5 No put this into one of your equations 2 x + y = 8 10 + (-2) = 8 Check this with the other equation X–y=7 5 – (-2) = 7 So x = 5 and y = -2

Solving Simultaneous Equations • • • • Sometimes equations need to be multiplied first 3 x + 2 y = 16 X+y=7 We want to get rid of the y’s so we need to multiply the second equation by 2 3 x + 2 y = 16 2 x + 2 y = 14 Now we can subtract X=2 Now we put this value into one of our equations 3 x + 2 y = 16 6 + 2 y =16 2 y = 10 Y=5 So x = 2 and y = 5

Solving Simultaneous Equations • • • • • How would you solve 2 x + 3 y = 11 5 x + 4 y = 24 Multiply both equations 2 x + 3 y = 11 multiplied by 4 5 x + 4 y = 24 multiplied by 3 8 x + 12 y = 44 15 x + 12 y = 72 Now we can subtract 15 x + 12 y = 72 8 x + 12 y = 44 7 x = 28 X=4 Now place this value into our equation 2 x + 3 y = 11 8 + 3 y = 11 3 y = 3 Y=1 So x = 4 and y = 1

Solving Simultaneous Equations • • • • • Solve these simultaneous equations X=2+y 3 x + y =14 We have to rearrange these equations so that the number is on its own X–y=2 3 x + y =14 How do we get rid of the y? We add the two equations X–y=2 3 x + y =14 4 x = 16 X=4 Then we put this value into one of our equations X–y=2 4–y=2 Y=2 So x = 4 and y = 2

Inequalities These lines all follow rules This purple line is always on y = 3 This dark red line is always on X= -10 This red line is always on x=7

Inequalities • Find the region where x ≤ 2

Inequalities • Find the region where y < x + 2 This line is y= x + 2 Because y is less than, but not equal to this line it has to be drawn as a dotted line Less than means under the line This line is y=x

Inequalities • Find the region where • Y ≥ -3, x ≤ 2 and y < x x≤ 2 y<x Y ≥ -3 This is the cross over region