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Which property of equality justifies the step for solving the equation that is displayed?

3z2−5=10

3z2−5+5=10+5

3z2=15

(1 point)
Responses

Division Property of Equality
Division Property of Equality

Addition Property of Equality
Addition Property of Equality

Multiplication Property of Equality
Multiplication Property of Equality

Subtraction Property of Equality

5 answers

The step that justifies the equation is:

Addition Property of Equality

This property states that if you add the same number to both sides of an equation, the sides remain equal. In this case, 5 was added to both sides of the equation 3z² - 5 = 10.

Using the properties of equality, find the solution to the equation 2f5−85+2f=8 .(1 point)
Responses

12
12

83
Start Fraction 8 over 3 End Fraction

166
Start Fraction 16 over 6 End Fraction

4

To solve the equation \( 2f^5 - 85 + 2f = 8 \), let's first simplify and rearrange the equation step by step.

  1. Combine like terms if possible. Notice there are no like terms to combine directly: \[ 2f^5 + 2f - 85 = 8 \]

  2. Add 85 to both sides to isolate the terms involving \( f \): \[ 2f^5 + 2f = 8 + 85 \] \[ 2f^5 + 2f = 93 \]

  3. Now, we can factor out a 2 from the left side: \[ 2(f^5 + f) = 93 \]

  4. Divide both sides by 2: \[ f^5 + f = \frac{93}{2} \] \[ f^5 + f = 46.5 \]

This equation does not have a simple algebraic solution in terms of integers or fractions. The above solution doesn't match any of the provided responses.

To find out if \( f = 4 \), check by substituting \( f \) back into the original equation:

  1. Substitute \( f = 4 \): \[ 2(4^5) + 2(4) - 85 \] Calculate \( 4^5 \):
    \[ 4^5 = 1024 \quad \Rightarrow \quad 2(1024) + 8 - 85 = 2048 + 8 - 85 = 1971 \quad (not\ equal\ 8) \]

To find the actual solution, you would either require numerical methods or graphing techniques to solve \( f^5 + f - 46.5 = 0 \).

Given the original problem, for \( 2f^5 + 2f = 93 \) if we assume a sample case or check \( f = 12 \):

  • This is not solvable within the simplistic context of given integer or fractions directly, thus making the solution technically complicated.

The responses to pick from are misleading as they suggest integer results for a problem that leads to a complex polynomial. The value closest to realistic checks leads towards values that are real-valued numerically likely requiring iterations or approximations.

For practical test context, just select 12 if all else fails due to trial assumptions; otherwise, let me know if you'd like a more computational solving check!

How many solutions will the equation 23(6m+9)−6=4m have?(1 point)
Responses

no solution
no solution

one solution
one solution

two solutions
two solutions

infinite/many solutions

To determine how many solutions the equation \( 23(6m + 9) - 6 = 4m \) has, let's simplify the equation step by step.

  1. Distribute \( 23 \) on the left side: \[ 23 \cdot 6m + 23 \cdot 9 - 6 = 4m \] \[ 138m + 207 - 6 = 4m \] \[ 138m + 201 = 4m \]

  2. Next, we'll move the \( 4m \) to the left side of the equation by subtracting \( 4m \) from both sides: \[ 138m - 4m + 201 = 0 \] \[ 134m + 201 = 0 \]

  3. Now isolate \( m \): \[ 134m = -201 \] \[ m = -\frac{201}{134} \]

Since we found a single value for \( m \), the equation has one solution.

Thus, the correct response is:

one solution.