Intersection of lines - from a graph • Intersection of ......VCE Maths Methods - Simultaneous...
Transcript of Intersection of lines - from a graph • Intersection of ......VCE Maths Methods - Simultaneous...
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)