Problem 1 :
Solve :
7(2 + 5x) = 3x + 14
Problem 2 :
Solve :
36 - 7x = -7(x - 5)
Problem 3 :
Solve :
-4(x + 3) = -12 - 4x
Problem 4 :
Solve :
-(3 - 6x) = 6x + 5
Problem 5 :
Solve :
-3(x + 4) = 2x - 37
Problem 6 :
Solve :
2(1 - x) + 5x = 3(x + 1)
Problem 7 :
Solve :
(¹⁄₂)(8y - 6) = 5y - (y + 3)
Problem 8 :
Solve :
(⅓)(9 - 6x) = 5 - 2x
Problem 9 :
(1/3)(15 - 6x) = 5 - ax
If the linear equation above is an identity, what is the value of a ?
Problem 10 :
4x + 13 = 7(x - 2) + bx
If the linear equation above has no solution, what is the value of b ?
1. Answer :
7(2 + 5x) = 3x + 14
Use the Distributive Property.
14 + 35x = 3x + 14
Subtract 3x from each side.
14 + 32x = 14
Subtract 14 from both sides.
32x = 0
Divide both sides by 32.
x = 0
The variable x does not vanish in the last step. Therefore, the given equation has only one solution, that is 0.
2. Answer :
36 - 7x = -7(x - 5)
Use the Distributive Property.
36 - 7x = -7x + 35
Add 7x to both sides.
36 = 35
The above result is false. Because 36 is not equal to 35. Because the result we get at the last step is false, the given equation has no solution.
3. Answer :
-4(x + 3) = -12 - 4x
Use the Distributive Property.
-4x - 12 = -12 - 4x
Add 4x to both sides.
-12 = -12
The above result is true. Because the result we get at the last step is true, the given equation has infinitely has many solutions.
4. Answer :
-(3 - 6x) = 6x + 5
Use the Distributive Property.
-3 + 6x = 6x + 5
Subtract 6x from both sides.
-3 = 5
The above result is false. Because -3 is not equal to 5. Because the result we get at the last step is false, the given equation has no solution.
5. Answer :
-3(x + 4) = 2x - 37
Use the Distributive Property.
-3x - 12 = 2x - 37
Subtract 2x from both sides.
-5x - 12 = -37
Add 12 to both sides.
-5x = -25
Divide both sides.
x = 5
The variable x does not vanish in the last step. Therefore, the given equation has only one solution, that is 5.
6. Answer :
2(1 - x) + 5x = 3(x + 1)
Use the Distributive Property.
2 - 2x + 5x = 3x + 3
2 + 3x = 3x + 3
Subtract 3x from each side.
2 = 3
The above result is false. Because 2 is not equal to 3. Because the result we get at the last step is false, the given equation has no solution.
7. Answer :
(¹⁄₂)(8y - 6) = 5y - (y + 3)
Use the Distributive Property.
4y - 3 = 5y - y - 3
4y - 3 = 4y - 3
Subtract 4y from each side.
-3 = -3
The above result is true. Because the result we get at the last step is true, the given equation has infinitely has many solutions.
8. Answer :
(⅓)(9 - 6x) = 5 - 2x
Use the Distributive Property.
3 - 2x = 5 - 2x
Add 2x to each side.
3 = 5
The above result is true. Because the result we get at the last step is true, the given equation has infinitely has many solutions.
9. Answer :
(⅓)(15 - 6x) = 5 - ax
Use the Distributive Property.
5 - 2x = 5 - ax
Because the given equation is an identity, the coefficients of like terms on both sides must be equal.
That is, coefficients of 'x' terms on the left side and right side must be equal.
So, equate the coefficients of 'x'.
-2 = -a
Multiply each side by (-1).
2 = a
If the given linear equation is an identity, the value of a is 2.
10. Answer :
4x + 13 = 7(x - 2) + bx
Use the Distributive Property.
4x + 13 = 7x - 14 + bx
4x + 13 = bx + 7x - 14
4x + 13 = (b + 7)x - 14
If (b + 7) = 4, we have
4x + 13 = 4x - 14
Subtract 4x from each side.
13 = -14
The above result is false. Because 13 is not equal to -14.
If (b + 7) = 4, the result we get is false and the given equation has no solution.
Solve for b in 'b + 7 = 4'.
b + 7 = 4
Subtract 7 from each side.
b = -3
If the given linear equation has no solution, the value of b is -3.
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