# Equations in One Variable- Grade 6

Grade 6 examples and questions on equations and problems in one variable with detailed solutions, at the bottom of the page, and explanations are presented. If you find that some of the questions are challenging do not skip them, spend time on them and work in groups. We learn math by solving challenging questions.

1. Which of the following is an equation in one variable?
1. 2x + 2
2. x + 2 = 4
3. 8 = x
4. 4 + x / 3
5. (x - 8) / 3 = 9
6. (2x +7) / 3

2. Which of the following is not an equation?
1. 3x - 9
2. x/2 + 4 = 6
3. 12 - 7 = 5
4. 8 & x
5. 5 = x / 4

3. Which of the following values of x satifies the equation 2 x - 4 = 4 ?
1. x = 0
2. x = 4
3. x = 2
4. x = - 2

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

5. Solve the following equations:
1. x - 6 = 12
2. 3 = x + 3
3. 2 + x = 8
4. 2 x = 16
5. x / 3 = 5

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

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

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

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

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

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

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

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

14. 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

1. 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.

1. x + 2 = 4
2. 8 = x
3. (x - 8) / 3 = 9
2. Solution
According to the definition given in 1, the following are not equations from the given list.

1. 3x - 9
2. 8 & x
3. 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.

1. 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.
2. 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.
4. Solution
We substitute x by the given value and evaluate both sides of the equation x / 3 - 1 = 2 and then compare them.

1. 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.
2. 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.
3. 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.
4. 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.
5. Solution
1. Solve x - 6 = 12
add +6 to both sides
x - 6 + 6 = 12 + 6
Simplify
x = 18 , solution to the given equation.
2. 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.
3. 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.
4. 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.
5. 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.
6. Solution
We solve each pair and compare the solutions.

1. 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
2. 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.
3. 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.
4. 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.
7. 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.
8. 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.
9. 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.
10. 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.
11. 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.
12. 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.
13. 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.
14. 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.

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