Simultaneous Equations Solving Sim. Equations Graphically Graphs as Mathematical Models Solving Simple Sim. Equations by Substitution Solving Simple Sim. Equations by elimination 19-May-19

Starter Questions 19-May-19

Simultaneous Equations S5 Int2 Straight Lines Learning Intention Success Criteria To solve simultaneous equations using graphical methods. Interpret information from a line graph. Plot line equations on a graph. 3. Find the coordinates were 2 lines intersect ( meet) 19-May-19

(1,3) Q. Find the equation of each line. Q. Write down the coordinates were they meet. 19-May-19

Q. Find the equation of each line. Q. Write down the coordinates where they meet. (-0.5,-0.5) 19-May-19

Q. Plot the lines. (1,1) Q. Write down the coordinates where they meet. 19-May-19

Starter Questions S5 Int2 8cm 5cm 19-May-19

Simultaneous Equations Straight Lines Learning Intention Success Criteria To use graphical methods to solve real-life mathematical models Draw line graphs given a table of points. 2. Find the coordinates were 2 lines intersect ( meet) 19-May-19

We can use straight line theory to work out real-life problems especially useful when trying to work out hire charges. Q. I need to hire a car for a number of days. Below are the hire charges charges for two companies. Complete tables and plot values on the same graph. 160 180 200 180 240 300 19-May-19

Who should I hire the car from? Summarise data ! Who should I hire the car from? Total Cost £ Arnold Up to 2 days Swinton Over 2 days Arnold Swinton Days 19-May-19

Key steps 1. Fill in tables 2. Plot points on the same graph ( pick scale carefully) 3. Identify intersection point ( where 2 lines meet) 4. Interpret graph information. 19-May-19

Starter Questions S5 Int2 19-May-19

Simultaneous Equations S5 Int2 Straight Lines Learning Intention Success Criteria To solve pairs of equations by substitution. 1. Apply the process of substitution to solve simple simultaneous equations. 19-May-19

Example 1 Solve the equations y = 2x y = x+1 by substitution 19-May-19

At the point of intersection y coordinates are equal: y = 2x y = x+1 so we have 2x = x+1 Rearranging we get : 2x - x = 1 x = 1 Finally : Sub into one of the equations to get y value y = 2x = 2 x 1 = 2 OR y = x+1 = 1 + 1 = 2 The solution is x = 1 y = 2 or (1,2) 19-May-19

y = x + 1 x + y = 4 by substitution (1.5, 2.5) Example 1 Solve the equations y = x + 1 x + y = 4 by substitution (1.5, 2.5) 19-May-19

The solution is x = 1.5 y = 2.5 (1.5,2.5) At the point of intersection y coordinates are equal: so we have x+1 = -x+4 y = x +1 y =-x+ 4 2x = 4 - 1 Rearranging we get : 2x = 3 x = 3 ÷ 2 = 1.5 Finally : Sub into one of the equations to get y value y = x +1 = 1.5 + 1 = 2.5 OR y = -x+4 = -1.5 + 4 = 2 .5 19-May-19

Starter Questions 19-May-19

Simultaneous Equations Straight Lines Learning Intention Success Criteria To solve simultaneous equations of 2 variables. Understand the term simultaneous equation. Understand the process for solving simultaneous equation of two variables. 3. Solve simple equations 19-May-19

Example 1 Solve the equations x + 2y = 14 x + y = 9 by elimination 19-May-19

Step 1: Label the equations x + 2y = 14 (1) x + y = 9 (2) Step 2: Decide what you want to eliminate Eliminate x by subtracting (2) from (1) x + 2y = 14 (1) x + y = 9 (2) y = 5 19-May-19

Substitute y = 5 in (2) x + y = 9 (2) x + 5 = 9 x = 9 - 5 Step 3: Sub into one of the equations to get other variable Substitute y = 5 in (2) x + y = 9 (2) x + 5 = 9 x = 9 - 5 The solution is x = 4 y = 5 x = 4 Step 4: Check answers by substituting into both equations x + 2y = 14 x + y = 9 ( 4 + 10 = 14) ( 4 + 5 = 9) 19-May-19

Example 2 Solve the equations 2x - y = 11 x - y = 4 by elimination 19-May-19

Step 1: Label the equations 2x - y = 11 (1) x - y = 4 (2) Step 2: Decide what you want to eliminate Eliminate y by subtracting (2) from (1) 2x - y = 11 (1) x - y = 4 (2) x = 7 19-May-19

Substitute x = 7 in (2) x - y = 4 (2) 7 - y = 4 y = 7 - 4 Step 3: Sub into one of the equations to get other variable Substitute x = 7 in (2) x - y = 4 (2) 7 - y = 4 y = 7 - 4 The solution is x =7 y =3 y = 3 Step 4: Check answers by substituting into both equations 2x - y = 11 x - y = 4 ( 14 - 3 = 11) ( 7 - 3 = 4) 19-May-19

Example 3 Solve the equations 2x - y = 6 x + y = 9 by elimination 19-May-19

Step 1: Label the equations 2x - y = 6 (1) x + y = 9 (2) Step 2: Decide what you want to eliminate Eliminate y by adding (1) and (2) 2x - y = 6 (1) x + y = 9 (2) 3x = 15 x = 15 ÷ 3 = 5 19-May-19

Substitute x = 5 in (2) x + y = 9 (2) 5 + y = 9 y = 9 - 5 Step 3: Sub into one of the equations to get other variable Substitute x = 5 in (2) x + y = 9 (2) 5 + y = 9 y = 9 - 5 The solution is x = 5 y = 4 y = 4 Step 4: Check answers by substituting into both equations 2x - y = 6 x + y = 9 ( 10 - 4 = 6) ( 5 + 4 = 9) 19-May-19

Starter Questions 19-May-19

Simultaneous Equations Straight Lines Learning Intention Success Criteria To solve harder simultaneous equations of 2 variables. 1. Apply the process for solving simultaneous equations to harder examples. 19-May-19

Example 1 Solve the equations 2x + y = 9 x - 3y = 1 by elimination 19-May-19

Step 1: Label the equations 2x + y = 9 (1) x -3y = 1 (2) Step 2: Decide what you want to eliminate Adding Eliminate y by : 2x + y = 9 x -3y = 1 (1) x3 6x + 3y = 27 (3) x - 3y = 1 (4) (2) x1 7x = 28 x = 28 ÷ 7 = 4 19-May-19

Substitute x = 4 in equation (1) Step 3: Sub into one of the equations to get other variable Substitute x = 4 in equation (1) 2 x 4 + y = 9 y = 9 – 8 y = 1 The solution is x = 4 y = 1 Step 4: Check answers by substituting into both equations 2x + y = 9 x -3y = 1 ( 8 + 1 = 9) ( 4 - 3 = 1) 19-May-19