Question

How many solutions does these equations have? One, infinite or no solutions?

1. 7(y+3)=5y+8

2. 10+6x=2(5+3x)

3. 3x+5-x=2x+7

4. 4x-4=2x+8

Answer 1

• 1. 7(y + 3) = 5y+8: one solution

• 2. 10 + 6x= 2(5+3x): infinite solutions

• 3. 3x + 5 - x=2x + 7: no solutions

• 4. 4x - 4 = 2x + 8: one solution.

Step-by-step explanation:

1. 7(y+3)=5y+8

a) Distributive property of multiplication over addtion:

• 7y + 21 = 5y + 8

b) Subtraction property of equalities:

• 7y - 5y = 8 - 21

c) Combine like terms:

• 2y = - 13

d) Division property of division:

• y = - 13/2

e) Conclusion: the equation has one solution.

2. 10 + 6x = 2(5+3x)

a) Distributive property of multiplication over addition:

• 10 + 6x = 10 + 6x

b) Subtraction property of equality:

• 6x - 6x = 10 - 10

c) Simplify (combine like terms)

• 0 = 0

That is an identity, i.e. that is always true, no matter the value of x.

d) Conclusion: the equation has infinite solutions.

3. 3x + 5 - x = 2x + 7

a) Combine like terms:

• 2x + 5 = 2x + 7

b) Subtraction property of equalities:

• 2x - 2x = 7 - 5

c) Combine like terms (simplify)

• 0 = 2

That is always false, no matter the values of x.

d) Conclusion: the equation does not have any solutions.

4. 4x - 4 = 2x + 8

a) Subtraction property of equality

• 4x - 2x - 4 = 2x - 2x + 8

b) Combine like terms

• 2x - 4 = 8

c) Addtiion property of equalities:

• 2x = 8 + 4

d) Combine like terms (simplify)

• 2x = 12

e) Division property of equalities:

• x = 6

f) Conclusion: the equation has one solution