2Tvwo 9.3 Intersecting lines (1)

• Wiskunde
• Middelbare school
• vwo
• Leerjaar 2

§9.3 Intersecting lines, ....

..... is today's hot dinner!

First a question on the ENTRANCE TICKET you just did.

This is how I did on the ET:

1. bad
2. good
3. moderate
4. excellent

9.3 Intersecting lines

When two lines intersect they meet in a point. 

Next slide shows this. The point of intersection (= p.o.i.) is  (4, 11)

That point has coordinates (x, y)

 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.

In the next slide ......

is shown how to put into practice what you learned in    

§9.1 and §9.2!

Solving linear equations turns out te be extremely useful

to calculate the point of intersection of 2 graphs!

How to CALCULATE the point of intersection of the graphs of two linear formulas ?

You can sometimes READ OFF the point of intersection. If you have no graph, you have to MAKE AN EQUATION  by making the two linear formulas equal to each other.

Solve the equation to find the x-value of the point of intersection.

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

you get this equation:                           1.5x + 5 = -x + 15

For the rest:  see next slide!

How to calculate the point of intersection of the graphs of two linear formulas ? You learnt this in §9.2 Solving equations!

you get this equation:                                                                  1.5x + 5 = -x + 15

adding 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 the coordinates of the point of intersection are:                       ( 4,11 )

You can check by filling x=4 into the other formula:

 y = -x + 15 = -4 + 15 = 11        gives the same answer, making us happy.

Let's do exercise 20 together!

+ open your Notebook and take a pen out of your pencilcase

+ Open your Textbook on p.97

+ Have a short look at 20, esp. the GRAPH!

+ Step by step try to do this exercise,

+ Answering all the questions  in your Notebook.

Solution:

20a y = -3x + 20 corresponds to

    graph 1, because this graph is falling and

    this formula has -3 as gradient.

    A negative gradient means a falling graph!

What equation??  Well, this one:

Make it in your Notebook, again.

Use all that you learned in §9.1 and §9.2 now!

Work it out in your Notebook right now!

The solution to the equation
5x + 2 = -3x + 20 is ........

​4​​1​​

​4​​1​​

1. 2
2. 4 1/2
3. - 2 1/41/4
4. 2 1/4

Solution equation:

This is how you are supposed 

to have worked out the equation in your

Notebook, just now.

Just like you did in §9.1 and §9.2!

What x-value?!

Well:  x = 2            

  

Work this out in your Notebook, now!      

​4​​1​​

Filling in x = 2 1/4 in y = 5x + 2
gives ...............

1. 14
2. 13 1/4
3. 13 3/4
4. 13

Solution:

So fill in x = 2 1/4 in this formula.

Work it out in your Notebook.

Fill in x = 2 1/4 in
y = -3x + 20

1. 13 1/4
2. 12 1/4
3. 12 3/4
4. 13.25

Solution:

exercise 20, once more, in full:

                     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 the 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 = 5 x 2.25 + 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 try it yourself with:

 exercises  21   t/m   24

Later this Lessonup will be carried on for a little while.

A little more about §9.3

On p.98 you are asked to

MAKE A SKETCH of the GRAPH of a FORMULA.

Making a GRAPH is a lot of work, 

making a SKETCH isn't! Let's look at it a bit closer.

WHY all this SKETCHING? You need this in §9.4 !

So LEARNING it NOW helps you doing the next paragraph.

When asked to MAKE a SKETCH of this FORMULA,

you might wonder:

- where does the sketch go through the VERTICAL AXIS and

- what is the SLOPE (helling) of it?

SOLUTION

SKETCH:  

Another example follows!  

MAKE a SKETCH for this FORMULA:

Again we wonder about:

- where does it go

   through the VERTICAL AXIS and

- what is the SLOPE?

MAKE a SKETCH for this FORMULA:

Theory on page 98

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

+  gradient tells you whether the line is 

a.  rising (if gradient is positive)

             or it is 

b.  falling (if gradient is negative)

Something about  24 ......

Something about  24 ......

HINTS to. 24a SKETCHING the graphs: 

+  you need to see where they intersect the 

         y-axis, and 

+  you need to see what the gradient is. 

Homework time

+ now do §9.3 

+ make:    21   t/m   24

+ send in slides of your work

+ enjoy your day!