Download Linear equations משוואות עם נעלמים = 1 variable:

Linear equations ‫=משוואות עם נעלמים‬
1 variable: - find the value of the variable that makes the equation true.
Rules for solving linear equation:
When you add or subtracted the same constant to both sides of the equation – the equality is
preserved. The new equation is equivalent to the original.
When both sides of an equation are multiplied or divided by the same nonzero constant – the
equality is preserved. The new equation is equivalent to the original.
Example: 3x-4 = 8
3x-4+4 = 8+4 (4 added to both sides)
3x=12
3x/3=12/3 (both sides divided by 3)
X=4
2 variables:
X+y = 16
5x-y = 2
Better to solve by add/subtract:
6x+ y-y = 18 6x = 18 x = 18/6 = 3
BACK TO THE 1ST ONE X+Y = 16, SO: Y = 13
Second degree – 1 variable: (always 2 solutions)
‫ריבועית משוואה‬:
X²+6X-9 = X+15
-X, -15 ‫יש להשוות את אחד מצדי המשוואה לאפס‬. ‫מבצעים את אותן פעולות בשני הצדדים‬:
X²+5X-24 = 0
Factoring ask “what 2 numbers that if you multiply them you get 5, and add them get -24”?
8, -3.  (X+8)(X-3) = 0
X+8=0 or: x-3=0.  x=-8, x=3.
ANOTHER WAY TO SOLVE: AX²+BX+C=0
FORMULA
=
=
INEQUALITIES ‫=משוואות עם נעלמים‬
-m, -10 -21 ≥ 6m /:6 
≥
= -8/3.

≥
Or:
-7m, +11-6m≥21 divide by -6 m≤WHEN DIVIDE OR MULTIPLY BY A NEGATIVE – YOU HAVE TO SWITCH THE SIGN (≥ TO ≤, ETC.)
QUANTITATIVE COMPARISON
HOW TO SOLVE?
‫ בשאלה שניתן לתנאי שעונה( אחת דוגמה מציבים‬-)‫ בעזרתו ופוסלים‬2 ‫תשובות‬.
Geometry
1 line going through 2 points.
Segment = AB ‫מקטע‬. ‫למשל‬
IF 2 LINES INTERSECT:
c
D
B
A
THE ANGELS THAT VERTICAL ARE
EQUAL
A=B, C=D, A+C=180°/ A FULL CIRCLE = 360°
TWO PERPENDICULAR LINES THAT INTERSECT: RIGHT ANGLE = 90°
PARALLEL: L3 ∏L4A=A, B=B.
4 of the angels are equal to each other
Coordinate Geometry
X-axis – the horizontal number line
Y-axis – the vertical number line
To find a distance between 2 points:
Pythagoras Theorem:
B
A
B
A
A
B
A
construct a right triangle, and then apply the
B
Example: PQ =
= √56.25
Linear equation - y=mx+b
mthe slope of the line
TO FIND THE SLOPE OF THE LINE BETWEEN 2 POINTS:
= 7.5
. EXAMPLE: THE SLOPE OF THE LINE PASSING THROUGH POINTS (4, 1.5) AND (-2,-3) IS:
=
bthe Y intercepts
TO FIND THE Y INTERCEPT - B:
AFTER YOU FOUND THE SLOPE – CHOOSE A POINT ON THE LINE AND SUBSTITUTE THE X AND Y IN THE
LINEAR EQUATION.
POLYGONS:
TRIANGLE:
THE BASIC POLYGON. THE MOST IMPORTANT GEOMETRY FACT: A+B+C = 180° (ANGLES).
‫הזוית הגדולה נמצאת מול הצלע הגדולה‬.
RULE: TRIANGLE CAN ONLY BE TRIANGLE IF THE SUM OF 2 SIDES IS BIGGER THAN THE 3RDSIDE.
EXAMPLE: WHICH OF THE FOLLOWING COULD BE THE LENGTHS OF A TRIANGLE?
1) 14, 17,2
2) 11,15,27
3) 13,13,0.028
ONLY 1 AND 3.
ISOSCELES – ‫ ְמשֻׁ לָּ ש ְשוֵה שוקיים‬. TWO SIDES ARE EQUAL, 2 ANGLES ARE EQUAL.
EQUILATERAL = ‫ ְמשֻׁ לָּ ש ְשוֵה צְ לָּ עֹות‬. ALL THE ANGLES = 60°, ALL THE SIDES ARE EQUAL.
RIGHT TRIANGLE = ANGLE OF 90°. IT MUST BE THE BIGGEST ANGLE.
Hypotenuse = ‫ הצלע שמול הזוית הגדולה במשולש ישר זוית‬.‫י ֶֶתר‬
Rule for any right triangle: a²+b²=c²
‫גדלים שחוזרים על עצמם במשולשים‬:
3:4:5 and any multiple of them (9:16:25…)
5:12:13
8:15:17
7:24:25
* If you take a square and divide it to 2 - you get this special triangle:
1:1:√21:1:1.4≈
IF YOU SEE A 45° ANGLE IT’S PROBABLY THAT.
* If you take an altitude in Equilateral triangle – you get this special triangle:
1: √3:2 1: 1.7≈:2
ANGLES- 30:60:90
222
222
AREA OF TRIANGLE: A = ½ *B*H.
(b=base, h = height).
122
122
The sum of the measures of the interior angles of an n-sided polygon is:
(n-2)*(180°)example: the sum for a hexagon (n=6) is 4*180° = 720°
REGULAR POLYGON = A POLYGON WITH ALL SIDES THE SAME LENGTH AND THE MEASURES OF ALL
INTERIOR ANGLES EQUAL.
QUADRILATERALS = ‫ְמרֻׁ בָּ עים‬
Every quadrilateral has four sides and 4 interior angles whose measures sum to 360°.
Area of a quadrilaterals = base*height
Special quadrilaterals:
Rectangle = a quadrilateral with all interior angles of 90°. Opposite sides are parallel and
have equal length, two diagonals have equal length.
Square = a rectangle with all sides of equal length.
Parallelogram = a quadrilateral with both pairs of opposite sides parallel. Opposite sides have
equal length; opposite interior angles have equal measure.
Trapezoid = a quadrilateral with one pair of opposite sides parallel.
Area of a Trapezoid = ½ (b1+b2)*(h)
CIRCLES
R = THE RADIUS
CIRCUMFERENCE = 2
AREA =
Arc = set of all points between 2 given points, can be measured in degrees. To find the length
of an arc it is important to know the ratio of arc length to circumference is equal to the ratio of
arc measure to 360°. Example – in a circle that circumference = 10
the arc interior angle
= 50°, so:
=
 ABC = (
)*(10
3 Dimensional Figures
Rectangular solid = has 6 rectangular surfaces called faces. The dimensions of rectangular
solid are length (l), width (w) and height (h).
Cube = a rectangular solid with l=w=h. Volume = l*w*h.
Surface area of a rectangular solid is the sum of the areas of the 6 facesA = 2(wl+lh+wh)
CYLINDER = VOLUME =
SURFACE AREA = SUM OF THE 2 BASE AREAS AND THE AREA OF THE CURVED SURFACE
2(