VCE Maths Methods - Simultaneous equations

Simultaneous equations

• Intersection of lines - from a graph

• Intersection of lines - substitution of equations

• Simultaneous equations - elimination method

• Simultaneous equations - matrix method

• Families of parallel lines - no solutions

• In!nite solutions

• Example question

VCE Maths Methods - Simultaneous equations

Intersection of lines - from a graph

y = x -1y = 2x -4

(3,2)

VCE Maths Methods - Simultaneous equations

Intersection of lines - substitution of equations

• The intersection of two lines can be found by solving simultaneous equations.

• At the point of intersection, both lines have the same x & y values.

• The method of substitution can be used to !nd where one equation is equal to another.

• eg y = 2x - 4 and y = x - 1

2x−4= x−1

2x−x =−1+4

x =3

To !nd the y value - substitute this x value into either equation.

y =3−1

y =2

VCE Maths Methods - Simultaneous equations

Simultaneous equations - elimination method

• If the equations are given in intercept form, it is easier to use the elimination method to solve.

• Both equations should be lined up together & one variable eliminated by adding or subtracting multiples of the equations.

• eg 7x - 11y = -13 and x + y = 11

7x−11y =−13 x+ y =11

7x−11y =−13

7x+7 y =77Multiply by 7 to get 7x

in both equations

7x−7x−11y −7 y =−13−77 Subtract boom equationfrom the top one to cancel x

−18 y =−90

y =5 Simplify & solve

x+5=11 Find x by substituting into either equation x =6

VCE Maths Methods - Simultaneous equations

Simultaneous equations - elimination method

7x - 11y = -13x + y = 11

(6,5)

VCE Maths Methods - Simultaneous equations

Simultaneous equations - matrix method

• Matrices can be used to solve a system of a number linear equations.

• The number of equations needed is equal to the number of variables in the equations.

• Here, two equations are used to solve for two variables, resulting in a 2x2 matrix.

• eg 7x - 11y = -13 and x + y = 11

7 −111 1

⎡

⎣⎢

⎤

⎦⎥

xy

⎡

⎣⎢

⎤

⎦⎥=

−1311

⎡

⎣⎢

⎤

⎦⎥

VCE Maths Methods - Simultaneous equations

Simultaneous equations - matrix method

7 −111 1

⎡

⎣⎢

⎤

⎦⎥

xy

⎡

⎣⎢

⎤

⎦⎥=

−1311

⎡

⎣⎢

⎤

⎦⎥

Create a 2 x 2 matrix of the multiples

y =5

Multiply matrices

x =6

7 −111 1

⎡

⎣⎢

⎤

⎦⎥

−1−1311

⎡

⎣⎢

⎤

⎦⎥=

xy

⎡

⎣⎢

⎤

⎦⎥

17−−11

1 11−1 7

⎡

⎣⎢

⎤

⎦⎥

−1311

⎡

⎣⎢

⎤

⎦⎥=

xy

⎡

⎣⎢

⎤

⎦⎥

118

−13+12113+77

⎡

⎣⎢

⎤

⎦⎥=

xy

⎡

⎣⎢

⎤

⎦⎥

xy

⎡

⎣⎢

⎤

⎦⎥=

118

10890

⎡

⎣⎢

⎤

⎦⎥=

65

⎡

⎣⎢

⎤

⎦⎥

Multiply by the inverse matrix

VCE Maths Methods - Simultaneous equations

Families of parallel lines - no solutions

• Two equations that represent parallel lines will have no solutions.

x +3y =6

4x +12y =36

1 34 12

⎡

⎣⎢

⎤

⎦⎥

Determinant = 1×12−3×4( )=0

x +3y =6

x +3y =9

Using the matrix method

The matrix can’t be inverted.

No solution to the equation can be found.

÷4

VCE Maths Methods - Simultaneous equations

Families of parallel lines - no solutions

x +3y =9

x +3y =6

VCE Maths Methods - Simultaneous equations

In!nite solutions

• Two equations that represent the same line will have an in!nite number of solutions.

• These two equations have the same gradient and y intercept.

• Both equations are multiples of each other.

9x +12y =36

6x +8 y =24

y = 36−9x

12

y =24−6x

8

m = 9

12= 3

4

m = 6

8= 3

4 c =24

8=3

c = 36

12=3

÷ 3

3x +4 y =12

Gradients are equal y intercepts are equal

3x +4 y =12÷ 2

VCE Maths Methods - Simultaneous equations

Example question

Find the value of m for which the simultaneous equations shown have:a) no solutionb) in!nitely many solutionsc) one solution

1)3x +my =5

2)(m+2)x +5y =m

No solution: the lines are parallel.(Gradients are equal.)

1) y = 5−3x

m=− 3

mx + 5

m

2) y =− (m+2)x

5+m

5

− (m+2)

5=− 3

m

m(m+2)=15

m2 +2m−15=0

(m−3)(m+5)=0

m =3,m =−5

m ≠3,m ≠−5

One solution: the lines have different gradients.

In!nitely many solutions: the lines are parallel and have the

same y intercept.

5m= m

5

m2 =25

m = ±5

m =3

m =−5

VCE Maths Methods - Simultaneous equations

Example question

(m+2)x +5y =m

3x +my =5

m = 4 (One solution)

m = 3 (No solution) m = -5 (Infinite solutions)