1819 H2Y §9.3

• Wiskunde
• Middelbare school
• havo, vwo
• Leerjaar 2

4a + 5 = 7a - 4   (- 4a both sides)

5 = 3a - 4             (+ 4 both sides)

9 = 3a                    (÷ 3 both sides)

3 = a

a = 3

7a + 2 = 3a - 8   (- 3a both sides)

4a + 2 = -8          (-2 both sides)

4a = -10                (÷ 4 both sides)

a = -2.5

8a + 5 = 5a - 5   (- 3a both sides)

3a + 5 = -5           (-5 both sides)

3a = -10                 (÷ 3 both sides)

a=−3​3​​1​​

8a - 7 = 12 - 6a      (+ 6a both sides)

14a - 7 = 12             (+ 7 both sides)

14a = 19                    (÷ 14 both sides)

a=1​14​​5​​

Which exercises should we discuss on the board?

What are the coordinates of the point of intersection?

Finding the point of intersection

We can easily read off the point of intersection for the

graphs on the right, because the point of intersection is

a grid point.

What if the point of intersection is not visible or

not a grid point?

Finding the point of intersection

How can we find the point of intersection for

these graphs?

Remember that a point of intersection is

a point on both graphs that have the same

horizontal and the same vertical coordinate.

We can use an equation to find the horizontal

coordinate!

Finding the point of intersection

First we need a formula for each line.

The green line:       y-intercept = 1

                                      gradient = 1 ÷ 2 = 0.5

formula: y = 0.5x + 1

The blue line:         y-intercept = -4

                                     gradient = 4 ÷ 5 = 0.8

formula: y = 0.8x - 4

Finding the point of intersection

 y = 0.5x + 1                     y = 0.8x - 4

As the graphs should both have the same y-value in the

point of intersection, we can make the equation:            

                                                      0.5x + 1 = 0.8x - 4

Solve this:                                1 = 0.3x - 4

                                                      5 = 0.3x

                                                      x =

This means the horizontal coordinate of the

point of intersection is

16​3​​2​​

16​3​​2​​

Finding the point of intersection

 y = 0.5x + 1                     y = 0.8x - 4

The horizontal coordinate of the point of intersection is

To find the vertical coordinate we fill in x =          

in each formula

So the coordinates of the point of intersection are

16​3​​2​​

16​3​​2​​

y=0.5⋅16​3​​2​​+1=9​3​​1​​

y=0.8⋅16​3​​2​​−4=9​3​​1​​

(16​3​​2​​,9​3​​1​​)

Finding the point of intersection

1) Make an equation by making both formulas equal to each other.

2) Calculate the horizontal coordinate by solving the equation.

3) Calculate the vertical coordinate  by filling in the horizontal coordinate in both formulas

4) Write down the point of intersection (.... , ....)

Sketching graphs

If you want to plot a graph, you first make a table, then a coordinate system and your graph should be very precise.

If you sketch a graph, your graph does not have to be very precise. It should show where it intersects the vertical axis, and if the graph is rising or falling.

Sketches can be useful for getting an idea about where the point of intersection is and when solving inequalities (§9.5).

Example

Sketch the graphs of the

formulas y = 2x + 5 and y = 3x - 1

in one coordinate system.

- both rising

(blue faster than green)

- y-intercepts 5 and -1

HW Friday

Make and correct §9.3