

Which of the following is an equation in one variable?
 2x + 2
 x + 2 = 4
 8 = x
 4 + x / 3
 (x  8) / 3 = 9
 (2x +7) / 3

Which of the following is not an equation?
 3x  9
 x/2 + 4 = 6
 12  7 = 5
 8 & x
 5 = x / 4

Which of the following values of x satifies the equation 2 x  4 = 4 ?
 x = 0
 x = 4
 x = 2
 x =  2

Which of the following values of x satifies the equation x / 3  1 = 2 ?
 x =  3
 x = 6
 x =  9
 x = 9

Solve the following equations:
 x  6 = 12
 3 = x + 3
 2 + x = 8
 2 x = 16
 x / 3 = 5

Which of the following pairs of equations have the same solution?
 x = 2 and 2 x = 4
 x + 3 = 6 and x + 4 = 8
 x / 2 = 2 and x =  4
 3 x = 9 and x + 1 = 4

What value of x makes the expression 2x + 6 equal to 12?

For what value of x the expressions 4 x + 6 and 2 + 12 have equal values?

The sum of d and 23 is 56. What is that value of d?

Seven subtracted from x is 41. What is that value of x?

The product of y and 6 is 36. What is the value of y?

The division of b by 5 is 4. What is the value of b?

Jacky has x cards and Jimmy has 23 cards. Together they have 121 cards. How many cards does Jacky have?

Jimmy, Toby and Dina contributed a total of $123 to buy a gift for their mother. Of the $123, Jimmy contributed $34 and Dina $45. How much money did Toby contribute for the gift?
Solutions to the Above Questions and Problems

Solution
An equation in mathematics is a statement that two mathematical expressions are equal. So equations must have an equal sign and mathematical expressions on each side. Hence only the following are equations from the given list.
 x + 2 = 4
 8 = x
 (x  8) / 3 = 9

Solution
According to the definition given in 1, the following are not equations from the given list.
 3x  9
 8 & x

Solution
We need to substitute x by the given value and evaluate both sides of the equation 2 x  4 = 4 and then compare them.
 x = 0
left side: 2 x  4 = 2(0)  4 = 0  4 =  4
right side: 4
The two sides are not equal and therefore x = 0 does not satisfy the given equation.
 x = 4
left side: 2 x  4 = 2(4)  4 = 8  4 = 4 right side: 4 The two sides are equal and therefore x = 4 satisfies the given equation and it is called a solution to the equation.
Only one value of x may satisfy the given equation and therefore there is no need to check the remaining values of x , they will not satisfy the given equation.

Solution
We substitute x by the given value and evaluate both sides of the equation x / 3  1 = 2 and then compare them.
 x =  3
left side: x / 3  1 = 2 = (3) / 3  1 =  1  1 =  2
right side: 2
The two sides are not equal and therefore x =  3 does not satisfy the given equation.
 x = 6
left side: x / 3  1 = 6 / 3  1 = 2  1 = 1 right side: 2 The two sides are not equal and therefore x = 6 does not satisfy the given equation.
 x =  9
left side: x / 3  1 = ( 9) / 3  1 =  3  1 =  4 right side: 2 The two sides are not equal and so x =  9 does not satisfy the given equation.
 x = 9
left side: x / 3  1 = (9) / 3  1 = 3  1 = 2 right side: 2 The two sides are equal and so x = 9 satisfies the given equation and is a solution.

Solution
 Solve x  6 = 12
add +6 to both sides
x  6 + 6 = 12 + 6
Simplify
x = 18 , solution to the given equation.
 Solve 3 = x + 3
subtract 3 from both sides of the equation
3  3 = x + 3  3
Simplify
0 = x , solution to the given equation.
 Solve 2 + x = 8
subtract 2 from both sides of the equation
2 + x  2 = 8  2
Simplify
x = 6 , solution to the given equation.
 Solve 2 x = 16
Divide both sides of the equation by 2
2 x / 2 = 16 / 2
Simplify
x = 8 , solution to the given equation.
 Solve x / 3 = 5
Multiply both sides of the equation by 3
3 (x / 3) = 3 (5)
Simplify
x = 15 , solution to the given equation.

Solution
We solve each pair and compare the solutions.
 Solve the two equations x = 2 and 2 x = 4
First equation: x = 2 is solved
Second Equation: Divide both sides of the equation by 2 and simplify
2 x / 2 = 4 / 2 gives x = 2
The two equations have the same solutions
 Solve the two equations x + 3 = 6 and x + 4 = 8
First equation: x + 3 = 6 ; subtract 3 from both sides and simplify
x + 3  3 = 6  3 gives x = 3
Second Equation: x + 4 = 8 ; subtract 4 from both sides and simplify
x + 4  4 = 8  4 gives x = 4
The two equations do not have the same solutions.
 Solve the two equations x / 2 = 2 and x =  4
First equation: x / 2 = 2 ; multiply by both sides by 2 and simplify
2(x / 2) = 2(2) gives x = 4
Second Equation: x =  4 is already solved
The two equations do not have the same solutions.
 Solve the two equations 3 x = 9 and x + 1 = 4
First equation: 3 x = 9 ; divide by both sides by 3 and simplify
3 x / 3 = 9 / 3 gives x = 3
Second Equation: x + 1 = 4 ; subtract 1 from both sides and simplify
x + 1  1 = 4  1 gives x = 3
The two equations have the same solutions.

Solution
The value of x that makes 2 x + 6 equal to 12 is the solution to the equation
2 x + 6 = 12
Subtract 6 to both sides and simplify
2 x + 6  6 = 12  6
2x = 6
Divide both sides of the equation by 2 and simplify
2 x / 2 = 6 / 2 gives x = 3
Check by substituting x by 3 in the given expression
2 x + 6 = 2 (3) + 6 = 6 + 6 = 12 which is equal to 12.
x = 3 makes 2 x + 6 equal to 12.

Solution
The value of x that makes 4 x + 6 equal to 2 + 12 is the solution to the equation
4 x + 6 = 2 + 12
Simplify the right side
4 x + 6 = 14
Subtract 6 to both sides and simplify
4 x + 6  6 = 14  6
4x = 8
Divide both sides by 4 and simplify
4 x / 4 = 8 / 4 gives x = 2
Check by substituting x by 2 in the expression 4 x + 6
4 x + 6 = 4 (2) + 6 = 8 + 6 = 14 which is equal to right side 2 + 12.
x = 2 makes 4 x + 6 equal to 2 + 12.

Solution
The sum is represented by + operation in math. Hence the phrase "The sum of d and 23" is represented by
d + 23
and "is 56" means is equal to 56. Hence the statement "the sum of d and 23 is 56" is represented by the equation
d + 23 = 56
and to find d, we need to solve the equation above. Subtract 23 from both sides of the equation
d + 23  23 = 56  23
Simplify and solve for d
d = 33
Check the answer to the question.
d + 23 = 33 + 23 = 56
"The sum of d (= 33) and 23 is 56" is correct.

Solution
The phrase "Seven subtracted from x" is represented by
x  7
and "is 41" means is equal to 41. Hence the statement "Seven subtracted from x is 41" is represented mathematically by the equation
x  7 = 41
We find x by solving the equation above. Add 7 to both sides of the equation
x  7 + 7 = 41 + 7
Simplify and solve for x
x = 48
Check the answer to the question.
48  7 = 41
"Seven subtracted from x( = 48) is 41" is correct.

Solution
The phrase "The product of y and 6" is represented by
6 × y = 6 y
and "is 36" means is equal to 36. Hence the statement "The product of y and 6 is 36" is represented mathematically by the equation
6 y = 36
y is found by solving the equation above. Divide both sides of the equation by 6.
6 y / 6 = 36 / 6
Simplify and solve for y
y = 6
Check the answer to the question.
6 × 6 = 36
"The product of y ( = 6) and 6 is 36" is correct.

Solution
The phrase "division of b by 5" is represented by
b / 5
and "is 4" means is equal to 4. Hence the statement "The division of b by 5 is 4" is represented mathematically by the equation
b / 5 = 4
Multiply both sides of the equation by 5.
5 (b / 5) = 5 × 4
Simplify and solve for b
b = 20
Check the answer to the question.
b / 5 = 20 / 5 = 4
"The division of b( = 20) by 5 is 4" is correct.

Solution
Jacky has x cards and Jimmy has 23 cards.
Jacky: x cards
Jimmy: 23 cards
and "together they have 121 cards" means the total (sum) number of cards of both. Hence the two statements "Jacky has x cards and Jimmy has 23 cards" and "together they have 121 cards" is represented mathematically by the equation
x + 23 = 121
Subtract 23 from both sides of the equation above.
x + 23  23 = 121  23
Simplify and solve for x
x = 98
Jacky has x cards which was found to be equal to 98
Check the answer to the problem by Jacky's and Jimmy's cards
98 + 23 = 121
The answer x = 98 is correct because when the cards are added they give a total of 121.

Solution
The gift bought cost $123 which is the total contributions of all three. Hence
Total : 123
We know what Jimmy and Dina contributed.
Jimmy : $34
Dina : $45
We do not know what Toby contributed and therefore this is the unknown in this problem. Hence let us give it a name using the letter c, for example, to the amount contributed by Toby.
Toby : c
They put all their money together to buy the gift. Hence the contributions of all three is represented by the sum
34 + 45 + c
All the money put togther was used to buy the gift which we know its cost $123; hence the equation
34 + 45 + c = 123
Simplify the left side and rewrite the equation as
c + 79 = 123
Subtract 79 from both sides of the equation.
c + 79  79 = 123  79
Simplify and solve.
c = 44 which is the contribution, in dollars, of Topy for the gift to his mother.
Check the answer to the problem by adding all contributions.
44 + 34 + 45 = 123
The answer c = 44 is correct because when all contributions are added they give a total of 123 which is the price paid for the gift.
