Download (direct link):
2x — 3 = 8
2x = 11 Adding 3 to both sides of the equation
x = 5.5 Dividing both sides of equation by 2
This alternative setting out is not acceptable:
2x — 3 = 8=2.x = ll=a; = 5.5
The following setting out is also unacceptable when an equals sign is used to start each line of working:
2x — 3 = 8
= 2x = 11 Adding 3 to both sides of the equation
— x =5.5 Dividing both sides of the equation by 2
References: Balancing an Equation, Graphs, Inequations, Linear Equation, Quadratic Equations, Solving an Equation.
In mathematics, the prefix equi means equal. Therefore an equiangular polygon means a polygon with all its angles equal in size. If a polygon is equiangular and all its sides are equal in length, it is called a regular polygon. An equiangular triangle is a triangle that has all its angles equal in size. Since the three angles add up to 180°, each angle is equal to 180 3- 3 = 60°. Such a triangle is usually called equilateral.
References: Equilateral Triangle, Polygon, Regular Polygon, Square.
If two or more distances are equidistant, they are equal.
References: Angle Bisector, Locus, Perpendicular Bisector.
174 EQUIVALENT EQUATIONS
This is a triangle with all three of its angles equal in size to 60° and all three of its sides equal in length (see figure a). An equilateral triangle has three axes of symmetry, and the order of rotational symmetry is three.
Example. Fold a piece of paper to make an equilateral triangle.
Solution. Use a circular piece of paper. The paper folding process is easier when the circle is cut out. Fold the paper in half to form a semicircle, and crease it (see figure b). Fold into another semicircle, and crease the paper. Where the two creases meet is the center of the circle. Make a mark at the center and call it point 0.
Now make a fold of the circle so that a point on the circumference of the circle just meets the newly found center of the circle 0, and crease it. That crease is one side of the equilateral triangle. Two more creases that are similar are made to form an equilateral triangle.
References: Equiangular, Isometric, Symmetry.
These are the steps used in solving equations, as set out in the following example.
Example. Solve the equation 3(x + 2) = 1 — x, showing three equivalent equations as steps of working.
EQUIVALENT FRACTIONS 175
3(x + 2) = I — x
3x + 6 = 1 — x Expanding the brackets
4x = — 5 Adding x and —6 to both sides of the equation
x = —1.25 Dividing both sides of the equation by 4
This value for x is the solution of the equation.
In algebra two expressions are equivalent if they contain the same information, but expressed in different forms. Equivalent expressions are expressions that are equal to each other. For example, x + 2x and 3x — x are equivalent expressions. 2x(x + 3) is an equivalent expression to 2x2 + 6x, and is obtained by expanding the brackets.
Fractions are equivalent when they have the same value, but are written in different ways. A convenient way of introducing equivalent fractions is to use areas. Suppose a rectangle has a length of 6 units and a width of 2 units, and is divided up in three different ways.
The rectangle in figure a is divided into 3 equal regions and 1 of them is shaded. This means that | of the whole rectangle is shaded. The rectangle in figure b is divided into 6 equal regions and 2 of them are shaded. In this case | of the whole rectangle is shaded. In the situation in figure c the same rectangle is divided into 12 equal regions and 4 of them are shaded. We now have ^ of the whole rectangle shaded.
Summary. Since equal areas are shaded in each rectangle, it means that the three fractions |, |, and ^ are equal, and therefore have the same value:
1 _ 2 _ 4_
3 — 6 — 12
When fractions have the same value we say they are equivalent, and the fraction j is the simplest form of these three equivalent fractions. Another process for finding equivalent fractions is given in the following example.
Example 1. Write down four fractions equivalent to |.
Solution. Multiply | by |, by |, by |, and by | to obtain the four equivalent fractions:
Therefore, four equivalent fractions to | are |, |, and j|. Equivalent fractions are used to solve problems.
Example 2. Find* if | = y.
Solution. It can be seen that 2x8 = 16,
Therefore x = 24.
Reference: Canceling Fractions.
This is an algorithm for finding prime numbers less than a certain number.
Example. Find all the prime numbers less than or equal to 48.
Solution. Write down, in order, a list of all the counting numbers from 1 to 48. Strike out every second number after 2. This eliminates even numbers. All even numbers, except 2, are not prime. Using the numbers 1-48 again, now strike out every third number, except 3. These are multiples of 3 and are not prime. Repeat the process by striking out every fourth, then every fifth, number, and that will be enough for the prime numbers up to 48. In short, the process is to strike out every multiple of 2, 3, 4, and 5, but not the first number in each case. It should be remembered that 1 is not a prime number and needs to be struck out. As the process develops you will find that some numbers have already been eliminated. This process isolates all the prime numbers, because they are the ones left at the end, just as large stones are left behind in a sieve: