Simultaneous Equations

S5 Int2 Simultaneous Equations Solving Sim. Equations Graphically Graphs as Mathematical Models Solving Simple Sim. Equations by Substitution www.mathsrevision.com Solving Simple Sim. Equations by elimination Solving harder type Sim. equations Created by Mr. Lafferty Maths Department

S5 Int2 Starter Questions www.mathsrevision.com Created by Mr. Lafferty Maths Department

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. • Find the coordinates where • 2 lines intersect ( meet) www.mathsrevision.com Created by Mr. Lafferty Maths Department

Q. Find the equation of each line. (1,3) Q. Write down the coordinates where they meet. Created by Mr. Lafferty Maths Department

Q. Find the equation of each line. Q. Write down the coordinates where they meet. (-0.5,-0.5) Created by Mr. Lafferty Maths Department

Q. Plot the lines. (1,1) Q. Write down the coordinates where they meet. Created by Mr. Lafferty Maths Department

Now try Exercise 2 Ch7 (page 84 ) www.mathsrevision.com Created by Mr. Lafferty Maths Department

S5 Int2 Starter Questions 8cm www.mathsrevision.com 5cm Created by Mr. Lafferty Maths Department

Simultaneous Equations S5 Int2 Straight Lines Learning Intention Success Criteria • To use graphical methods to solve real-life mathematical models • Draw line graphs given a table of points. • Find the coordinates where • 2 lines intersect ( meet) www.mathsrevision.com Created by Mr. Lafferty Maths Department

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 Created by Mr. Lafferty Maths Department

Summarise data ! Who should I hire the car from? Arnold Total Cost £ Up to 2 days Swinton Over 2 days Arnold Swinton Days Created by Mr. Lafferty Maths Department

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. Created by Mr. Lafferty Maths Department

Now try Exercise 3 Ch7 (page 85 ) www.mathsrevision.com Created by Mr. Lafferty Maths Department

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. www.mathsrevision.com Created by Mr. Lafferty Maths Department

Example 1 Solve the equations y = 2x y = x+1 by substitution Created by Mr. Lafferty Maths Department

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

Example 1 Solve the equations y = x + 1 x + y = 4 by substitution (1.5, 2.5) Created by Mr. Lafferty Maths Department

y = x +1 y =-x+ 4 The solution is x = 1.5 y = 2.5 (1.5,2.5) At the point of intersection y coordinates are equal: x+1 = -x+4 so we have 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 y = -x+4 = -1.5 + 4 = 2 .5 OR Created by Mr. Lafferty Maths Department

Now try Ex 4 Ch7 (page88 ) www.mathsrevision.com Created by Mr. Lafferty Maths Department

Simultaneous Equations S5 Int2 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 www.mathsrevision.com Created by Mr. Lafferty Maths Department

Example 1 Solve the equations x + 2y = 14 x + y = 9 by elimination Created by Mr. Lafferty Maths Department

Step 1: Label the equations x + 2y = 14 (A) x + y = 9 (B) Step 2: Decide what you want to eliminate Eliminate x by subtracting (B) from (A) x + 2y = 14 (A) x + y = 9 (B) y = 5 Created by Mr. Lafferty Maths Department

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

Example 2 Solve the equations 2x - y = 11 x - y = 4 by elimination Created by Mr. Lafferty Maths Department

Step 1: Label the equations 2x - y = 11 (A) x - y = 4 (B) Step 2: Decide what you want to eliminate Eliminate y by subtracting (B) from (A) 2x - y = 11 (A) x - y = 4 (B) x = 7 Created by Mr. Lafferty Maths Department

Step 3: Sub into one of the equations to get other variable Substitute x = 7 in (B) x - y = 4 (B) 7 - y = 4 y = 7 - 4 The solution is x =7 y =3 y = 3 Step 4: Check answers by substituting into both equations ( 14 - 3 = 11) 2x - y = 11 x - y = 4 ( 7 - 3 = 4) Created by Mr. Lafferty Maths Department

Example 3 Solve the equations 2x - y = 6 x + y = 9 by elimination Created by Mr. Lafferty Maths Department

Step 1: Label the equations 2x - y = 6 (A) x + y = 9 (B) Step 2: Decide what you want to eliminate Eliminate y by adding (A) from (B) 2x - y = 6 (A) x + y = 9 (B) x = 15 ÷ 3 = 5 3x = 15 Created by Mr. Lafferty Maths Department

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

Now try Ex 5A Ch7 (page89 ) www.mathsrevision.com Created by Mr. Lafferty Maths Department

Simultaneous Equations S5 Int2 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. www.mathsrevision.com Created by Mr. Lafferty Maths Department

Example 1 Solve the equations 2x + y = 9 x - 3y = 1 by elimination Created by Mr. Lafferty Maths Department

Step 1: Label the equations 2x + y = 9 (A) x -3y = 1 (B) Step 2: Decide what you want to eliminate Adding Eliminate y by : (A) x3 2x + y = 9 x -3y = 1 6x + 3y = 27 (C) x - 3y = 1(D) (B) x1 7x = 28 x = 28 ÷ 7 = 4 Created by Mr. Lafferty Maths Department

Step 3: Sub into one of the equations to get other variable Substitute x = 4 in equation (A) 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 ( 8 + 1 = 9) 2x + y = 9 x -3y = 1 ( 4 - 3 = 1) Created by Mr. Lafferty Maths Department

Example 2 Solve the equations 3x + 2y = 13 2x + y = 8 by elimination Created by Mr. Lafferty Maths Department

Step 1: Label the equations 3x + 2y = 13 (A) 2x + y = 8 (B) Step 2: Decide what you want to eliminate Subtract Eliminate y by : (A) x1 3x + 2y = 13 2x + y = 8 3x + 2y = 13 (C) 4x + 2y = 16(D) (B) x2 -x = -3 x = 3 Created by Mr. Lafferty Maths Department

Step 3: Sub into one of the equations to get other variable Substitute x = 3 in equation (B) 2 x 3 + y = 8 y = 8 – 6 y = 2 The solution is x = 3 y = 2 Step 4: Check answers by substituting into both equations ( 9 + 4 = 13) 3x + 2y = 13 2x + y = 8 ( 6 + 2 = 8) Created by Mr. Lafferty Maths Department

Now try Ex 5B Ch7 (page90 ) www.mathsrevision.com Created by Mr. Lafferty Maths Department

Simultaneous Equations