# 26-dec-15 solving sim. equations graphically solving simple sim. equations by substitution...

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*Solving Sim. Equations GraphicallySolving Simple Sim. Equations by SubstitutionSimultaneous Equationswww.mathsrevision.comSolving Simple Sim. Equations by eliminationSolving harder type Sim. equationsS5 Int2Graphs as Mathematical Models

* Starter Questionswww.mathsrevision.comS5 Int2

*www.mathsrevision.comLearning IntentionSuccess CriteriaTo solve simultaneous equations using graphical methods.Simultaneous EquationsInterpret information from a line graph.

Plot line equations on a graph.

3.Find the coordinates were 2 lines intersect ( meet)

Straight LinesS5 Int2

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

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

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*Q. Plot the lines.(1,1)Q. Write down the coordinates where they meet.

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Now try Exercise 2Ch7 (page 84 )

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* Starter Questionswww.mathsrevision.comS5 Int25cm8cm

*www.mathsrevision.comLearning IntentionSuccess CriteriaTo use graphical methods to solve real-life mathematical modelsSimultaneous EquationsDraw line graphs given a table of points.

2.Find the coordinates were 2 lines intersect ( meet)

Straight LinesS5 Int2

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*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.160180200180240300

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*DaysTotal Cost ArnoldSwintonSummarise data ! Who should I hire the car from?Up to 2 days SwintonOver 2 days Arnold

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*Key steps1. Fill in tables2. Plot points on the same graph ( pick scale carefully)3. Identify intersection point ( where 2 lines meet)4. Interpret graph information.

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Now try Exercise 3Ch7 (page 85 )

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* Starter Questionswww.mathsrevision.comS5 Int2

*www.mathsrevision.comLearning IntentionSuccess CriteriaTo solve pairs of equations by substitution.Simultaneous Equations1.Apply the process of substitution to solve simple simultaneous equations.Straight LinesS5 Int2

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*Example 1

Solve the equations

y = 2x y = x+1

by substitution

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*At the point of intersection y coordinates are equal:2x = x+1 Rearranging we get :2x - x = 1 x = 1 Finally : Sub into one of the equations to get y valuey = 2x = 2 x 1 = 2ORy = x+1 = 1 + 1 = 2 so we haveThe solution is x = 1 y = 2 or (1,2)

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*Example 1

Solve the equations

y = x + 1 x + y = 4

by substitution(1.5, 2.5)

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*At the point of intersection y coordinates are equal:x+1 = -x+4 Rearranging we get :2x = 4 - 12x = 3 Finally : Sub into one of the equations to get y valuey = x +1 = 1.5 + 1 = 2.5y = -x+4 = -1.5 + 4 = 2 .5so we haveThe solution is x = 1.5 y = 2.5 (1.5,2.5)x = 3 2 = 1.5 OR

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Now try Ex 4Ch7 (page88 )

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* Starter Questionswww.mathsrevision.comS5 Int2

*www.mathsrevision.comLearning IntentionSuccess CriteriaTo solve simultaneous equations of 2 variables.Simultaneous EquationsUnderstand the term simultaneous equation.

Understand the process for solving simultaneous equation of two variables.

3.Solve simple equationsStraight LinesS5 Int2

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*Example 1

Solve the equations

x + 2y = 14 x + y = 9

by elimination

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*Step 1: Label the equationsx + 2y = 14 (A)x + y = 9 (B) Step 2: Decide what you want to eliminateEliminate x by subtracting (B) from (A)x + 2y = 14 (A)x + y = 9 (B)y = 5

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

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*Example 2

Solve the equations

2x - y = 11 x - y = 4

by elimination

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*Step 1: Label the equations2x - y = 11 (A) x - y = 4 (B) Step 2: Decide what you want to eliminateEliminate y by subtracting (B) from (A)2x - y = 11 (A) x - y = 4 (B)x = 7

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*Step 3: Sub into one of the equations to get other variableSubstitute x = 7 in (B)x - y = 4 (B)7 - y = 4The solution is x =7 y =3Step 4:Check answers by substituting into both equationsy = 7 - 4y = 32x - y = 11 x - y = 4( 14 - 3 = 11)( 7 - 3 = 4)

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*Example 3

Solve the equations

2x - y = 6 x + y = 9

by elimination

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*Step 1: Label the equations2x - y = 6 (A) x + y = 9 (B) Step 2: Decide what you want to eliminateEliminate y by adding (A) from (B)2x - y = 6 (A) x + y = 9 (B)3x = 15x = 15 3 = 5

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*Step 3: Sub into one of the equations to get other variableSubstitute x = 5 in (B)x + y = 9 (B)5 + y = 9The solution is x = 5 y = 4Step 4:

Check answers by substituting into both equationsy = 9 - 5y = 42x - y = 6 x + y = 9( 10 - 4 = 6)( 5 + 4 = 9)

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Now try Ex 5ACh7 (page89 )

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* Starter Questionswww.mathsrevision.comS5 Int2

*www.mathsrevision.comLearning IntentionSuccess CriteriaTo solve harder simultaneous equations of 2 variables.Simultaneous Equations1.Apply the process for solving simultaneous equations to harder examples.Straight LinesS5 Int2

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*Example 1

Solve the equations

2x + y = 9 x - 3y = 1

by elimination

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*2x + y = 9 x -3y = 1Step 1: Label the equations2x + y = 9(A) x -3y = 1 (B) Step 2: Decide what you want to eliminateEliminate y by :7x = 286x + 3y = 27 (C) x - 3y = 1 (D) x = 28 7 = 4Adding(A) x3(B) x1

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*Step 3: Sub into one of the equations to get other variableSubstitute x = 4 in equation (A)2 x 4 + y = 9 y = 9 8 The solution is x = 4 y = 1Step 4:

Check answers by substituting into both equationsy = 12x + y = 9 x -3y = 1( 8 + 1 = 9)( 4 - 3 = 1)

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*Example 2

Solve the equations

3x + 2y = 132x + y = 8

by elimination

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*3x + 2y = 13 2x + y = 8Step 1: Label the equations3x + 2y = 13(A)2x + y = 8 (B) Step 2: Decide what you want to eliminateEliminate y by :-x = -33x + 2y = 13 (C) 4x + 2y = 16 (D) x = 3Subtract(A) x1(B) x2

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*Step 3: Sub into one of the equations to get other variableSubstitute x = 3 in equation (B)2 x 3 + y = 8 y = 8 6 The solution is x = 3 y = 2Step 4:

Check answers by substituting into both equationsy = 23x + 2y = 132x + y = 8( 9 + 4 = 13)( 6 + 2 = 8)

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Now try Ex 5BCh7 (page90 )

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