An equation is a mathematical sentence with an equal sign. To translate a word sentence into an equation, choose a variable to represent one of the unspecified numbers or measures in the sentence. This is called defining a variable. Then use the variable to write equations for the unspecified numbers.
In the following examples 1-4, translate each sentence into an equation.
Example 1 :
Twice a number increased by fourteen is identical to fifty.
Solution :
Let x be the number.
Equation :
2x + 14 = 15
Example 2 :
Half the sum of seven and a number is the same as the number decreased by two.
Solution :
Let y be the number.
Equation :
(1/2)(7 + y) = y - 2
Example 3 :
The quotient of m and n equals four more than one-third the sum of m and n.
Solution :
Equation :
m/n = (1/3)(m + n) + 4
Example 4 :
The cube of x plus the square of y is equal to fifty two.
Solution :
Equation :
x^{3} + y^{2} = 52
Consecutive Integers :
......., -3, -2, -1, 0, 1, 2, 3, .......
n, n + 1, n + 2 are three consecutive integers if n is an integer.
Consecutive Even Integers :
......., -6, -4, -2, 0, 2, 4, 6, .......
n, n + 2, n + 4 are three consecutive integers if n is an even integer.
Consecutive Odd Integers :
......., -5, -3, -1, 0, 1, 3, 5, .......
n, n + 2, n + 4 are three consecutive integers if n is an odd integer.
In the following examples 5-6, write an equation to represent the given relationship between integers.
Example 1 :
The sum of four consecutive integers is -54,
Solution :
Let x be the first integer.
Then, the remaining three integers are
x + 1, x + 2 and x + 3
Equation :
x + (x + 1) + (x + 2) + (x + 3) = -54
Example 2 :
The product of three consecutive odd integers is 693.
Solution :
Let y be the first integer.
Then, the remaining two integers are
y + 2 and y + 4
Equation :
y(y + 2)(y + 4) = 693
Example 3 :
8 less than the sum of two consecutive integers is equal to 55.
Solution :
Let x be the first integer.
Then, the second integer is x + 1.
Equation :
x + (x + 1) - 8 = 55
Example 4 :
5 more than the product of two consecutive integers is equal to 25.
Solution :
Let y be the first integer.
Then, the second integer is y + 1.
Equation :
y(y + 1) + 5 = 25
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