Starter Solve: a) 4x = -16 b) x + 5 = -6 c) 2x - 3 = 11 d) 8 – 6x = 26 Substitute for x = -1, y = 5 e) 3x + 2y f) 4y – 6x x = -4 x = -11 x = 7 x = -3 -3 + 10 = 7 20 + 6 = 26

+ + = £12 + = £9 How much do the burgers cost? How much do the chips cost? + + = £12 + = £9

+ + + = £12 + = £8 How much do the burgers cost? How much do the chips cost? + + + = £12 + = £8

+ + = £8 + = £5 How much do the burgers cost? How much do the chips cost? + + = £8 + = £5

Simultaneous Equations Scale up each term in one or both equations to make the coefficients the same for either the x terms or the y terms. Subtract if the signs in front of these are the same. Add if the signs in front of these are the different.

Scale up (if necessary) Add or subtract (to eliminate) 5x + y = 20 …(1) 2x + y = 11 …(2) Scale up (if necessary) - Add or subtract (to eliminate) 3x = 9 x = 3 Solve (to find x) 5x + y = 20 15 + y = 20 Substitute in (to find y) y = 5

Scale up (if necessary) Add or subtract (to eliminate) 7x + 2y = 32 …(1) 3x – 2y = 8 …(2) Scale up (if necessary) + Add or subtract (to eliminate) 10x = 40 x = 4 Solve (to find x) 7x + 2y = 32 28 + 2y = 32 Substitute in (to find y) 2y = 4 y = 2

+ + + + = £20 + = £8.50 How much do the burgers cost? How much do the chips cost? + + + + = £20 + = £8.50

Scale up (if necessary) Add or subtract (to eliminate) 12x – 2y = 8 …(1) 5x + y = 18 …(2) x1 Scale up (if necessary) x2 12x – 2y = 8 10x + 2y = 36 + Add or subtract (to eliminate) 22x = 44 Solve (to find x) x = 2 12x – 2y = 8 24 – 2y = 8 -2y = -16 Substitute in (to find y) y = 8

Scale up (if necessary) Add or subtract (to eliminate) 7x – 3y = 29 …(1) 2x + 5y = 20 …(2) x2 Scale up (if necessary) x7 14x – 6y = 58 14x + 35y = 140 - Add or subtract (to eliminate) -41y = -82 Solve (to find y) y = 2 7x – 3y = 29 7x – 6 = 29 7x = 35 Substitute in (to find x) x = 5

+ + + + = £20 + = £8.50 How much do the burgers cost? How much do the chips cost? + + + + = £20 + = £8.50

Answers 1) x = 2, y = 3 2) x = 2, y = 4 3) x = 4, y = -1 4) x = 6, y = -5 5) x = 4.5, y = -3 6) x = -3, y = 5 7) x = 3, y = -0.5 8) x = -2, y = -5

Show me a pair of simultaneous equations where x = 3 and y = 2

A cinema sells adult tickets and child tickets.   The total cost of 3 adult tickets and 1 child ticket is £30. The total cost of 1 adult ticket and 3 child tickets is £22. Work out the cost of an adult ticket and the cost of a child ticket.     adult ticket £............................................... child ticket £............................................... (Total for Question is 4 marks)

Copy and complete the following table: Starter Copy and complete the following table: Equation Gradient y – intercept y = 3x + 2 2 7 y = 3x 3 y = -4x + 5 ½ 4 2y = 4x + 6

Answers Equation Gradient y – intercept y = 3x + 2 3 2 y = 2x + 7 7 y = -4x + 5 -4 5 y = ½x + 4 ½ 4 2y = 4x + 6

We are learning to solve simultaneous equations graphically.

Remember: y = mx + c m is the gradient, or the slope of the graph c is the y-intercept, or where the graph cuts the y-axis

Solve the simultaneous equations y = 2x + 1 and y = 3 graphically: Start by sketching y = 2x + 1 Start at 1 on the y-axis. For every 1 across, go up 2. Join with a straight line.

Solve the simultaneous equations y = 2x + 1 and y = 3 graphically: Start by sketching y = 2x + 1 Start at 1 on the y-axis. For every 1 across, go up 2. Join with a straight line.

Solve the simultaneous equations y = 2x + 1 and y = 3 graphically: The solution is the coordinate where the graphs cross. (1, 3) So x = 1 and y = 3.

Solve the simultaneous equations y = 3x + 2 and y = 6 – x graphically: Start by sketching y = 3x + 2 Start at 2 on the y-axis. For every 1 across, go up 3. Join with a straight line.

Solve the simultaneous equations y = 3x + 2 and y = 6 – x graphically: Now sketch y = 6 – x. Start at 6 on the y-axis. For every 1 across, go down 1. Join with a straight line.

Solve the simultaneous equations y = 3x + 2 and y = 6 – x graphically: The solution is the coordinate where the graphs cross. (1, 5) So x = 1 and y = 5.

Show me a pair of simultaneous equations with a solution at (5, 2). True/Never/Sometimes: All linear graphs intersect.

Answers x = 4, y = 8 x = -2, y = -6 x = 1, y = 3 x = 4, y = 0

Starter Solve the following, giving you answers to 2 d.p. where necessary: a) x² - x – 12 = 0 b) 6x² - x – 15 = 0 c) 3x² + 2x – 9 = 0 d) 4x² - 6x – 2 = 0 x = -3, 4 x = 1.67, -1.5 x = 1.43, -2.10 x = 1.78, -0.28

Simultaneous Equations (where one is linear and one is a quadratic) Linear graph Quadratic graph Solutions

Simultaneous Equations (where one is linear and one is a quadratic) y = 2x² + 6x + 4 y = -9x - 24 If they are both equal to y, they are equal to each other. -9x - 24 = 2x² + 6x + 4 0 = 2x² + 15x + 28 Manipulate equation so it equals 0. 0 = (2x + 7)(x + 4) Solve to find x. x = -7/2 or -4 Substitute to find corresponding y values. y = -9x - 24 y = -9x - 24 y = 63/2 - 24 y = 36 - 24 Write as coordinates. y = 15/2 y = 12 (-7/2, 15/2) and (-4, 12)

Simultaneous Equations (where one is linear and one is a circle) Linear graph Circle Solutions

Simultaneous Equations (where one is linear and one is a circle) x² + y² = 16 y = 2x - 5 Substitute the second equation for y in the first equation. x² + (2x – 5)² = 16 x² + 4x² - 20x + 25 – 16 = 0 Expand the brackets and manipulate equation so it equals 0. 5x² - 20x + 9 = 0 x = 3.48 or 0.52 Solve to find x. y = 2x - 5 y = 2x - 5 Substitute to find corresponding y values. y = 2(3.48) - 5 y = 2(0.52) - 5 y = 1.96 y = -3.96 Write as coordinates. (3.48, 1.96) and (0.52, -3.96)

Answers (3.85, 9.85) (-5, -3) (-2.85, 3.15) (3, 5) (-1, 3) (4/3, 2/3) (0.2, 4.2) (1, 1)