2TTO 9.3 Intersecting lines (2)

• Wiskunde
• Middelbare school
• vwo
• Leerjaar 2

9.3 Intersecting lines

When two lines intersect they meet in a point.

That point has coordinates (x,y)

where x is the value on the horizontal axis

and y is the value on the vertical axis.

so 

 for this particular value of x both lines have the same value for y. You could say they are equal to each other.

How to calculate the point of intersection of the graphs of two linear formulas

You can also find the point of intersection by putting the two linear formulas equal to each other to make an equation.

Solve the eqaution to find the x-vale of the point of intersection.

e.g. with formulas y= 1.5x + 5 and y = -x + 15 

you get equation   1.5x + 5 = -x + 15

add 1x to both sides gives   2.5x + 5 = 15

subtract 5 from both sides gives 2.5x = 10 so x = 4

Now fill x= 4 into one of the formulas to calculate y

y = 1.5 x 4 + 5 = 6 + 5 = 11

so coordinates of point of intersection are (4,11)

You can check by filling x=4 into other formula y = -4 + 15 = 11 which gives same answer

exercise 20

                     y= 5x + 2 and y = -3x +20

a) graph 1 starts high and gets lower so y = -3x + 20

b) when they intersect they are equal so 5x + 2 = -3x + 20

c) solving equation  

    first add 3x to both sides giving 8x + 2 = 20

    then subtracting 2 from both sides   8x = 18

    then dividing both sides by 8  gives x = 2.25

d) fill x= 2.25 into y = 5x + 2  gives y = 5x2.25 + 2 = 11.25 + 2 = 13.25

e) fill x= 2.25 into y = -3x + 20 gives y = -3 x 2.25 + 20 = -6.75 + 20 = 13.25

f) yes, they both give y=13.25 when x= 2.25

Now it's your turn to try

Do exercise 23

ex 22

a) To calculate gradient of line l gradient = (35-20)/ (10-0) = 15/10 = 1.5 

     That is difference in y coordinates divided by difference in x coordinates

b) the y-intercept is value when x=0 so in this case (0,20) gives y-intercept = 20

c) formula is y = 1.5x + 20

d)Gradient for line m Gradient = (15-5)/ (5 - 0) = 10/5 = 2

   and  y-intercept is 5 (from (0,5)

   so formula is y = 2x + 5

e) 2x + 5 = 1.5x + 20

subtract 1.5x from both sides gives 0.5x +5 = 20

subtract 5 from both sides gives 0.5x = 15

divide by 0.5 gives x = 30 and filling this into formula gives y = 2 x 30 + 5 = 65 so (30, 65)

Theory on page 98

y-intercept tells you where the line intercepts the y-axis 

gradient tells you whether the line is rising (if gradient is positive)

or if it is falling (if gradient is negative)

Now try exercise 24

Hint to sketch the graphs you need to see where intersects the 

y-axis and you need to see what the gradient is. 

How much you add each time tells you how much it rises or how much you subtract each time tells you how much it falls.

Make a table for each formula using values x= 0 to  x= 3

Time for homework

9.3: 23, 24 and of page 116: E1 and E2