Simultaneous Equations

Question 1

When would you use solving by substitution?

Answer 1

When one or both equations are non-linear

eg eg
y=4x + 1 (1) y=x^2 + y^2 = 5 (1)
y=2x^2-3x+4 (2) y=2/x (2)
Sub (2) in (1) Sub (2) in (1)
2x^2-3x+4= 4x + 1 x^2 + (2/x)^2 =5

Question 2

Determine the number of times the line y=x-1 intersects the curve y=x^2-2x+6

Answer 2

For intersection sub (2) in (1)
x^2-2x+6 = x-1
x^2-3x+7=0
a=1 b=-3 c=7
sub into b^2-4ac
b^2-4ac=-19
so b^2-4ac<0
So no real roots, the line and curve don't intersect

Question 3

Step process of having to find if the lines intersect and how many times?

Answer 3

For intersection make the lines equal each other
Simplify so it is in the from of quadratic equation
From this you can find the values of a, b and c
So you can sub these values into discriminant
Answer will tell you how many solutions

Question 4

What does it mean if b^2-4ac<0 when you are trying to work out the number of times a line intersects a curve?

Answer 4

b^2-4ac<0
no real solutions
doesn’t intersect

Question 5

What does it mean if b^2-4ac=0 when you are trying to work out the number of times a line intersects a curve?

Answer 5

b^2-4ac=0
one solution
the line is a tangent to the curve

Question 6

What does it mean if b^2-4ac>0 when you are trying to work out the number of times a line intersects a curve?

Answer 6

two real roots

intersects twice

Question 7

What does repeated roots mean?

Answer 7

1 root

Question 8

What is the name of the geometric relationship between the line and the curve if b^2-4ac=0

Answer 8

The line is a tangent to the curve