Determine the missing values that would make this equation have zero solutions.

Question

3 (5x + 6) = 3x + [?]x + 18

Answer 1

To determine the missing value that would make the equation \(3 (5x + 6) = 3x + [?]x + 18\) have zero solutions, we first simplify both sides of the equation.

Starting with the left side:

\[ 3(5x + 6) = 15x + 18 \]

Now, let's rewrite the right side with the missing value represented as \(a\):

\[ 3x + ax + 18 \]

This can be combined to:

\[ (3 + a)x + 18 \]

Now, we set the two sides of the equation equal to each other:

\[ 15x + 18 = (3 + a)x + 18 \]

Next, we'll subtract \(18\) from both sides:

\[ 15x = (3 + a)x \]

To have zero solutions, the left side cannot equal the right side for any \(x\), which means the coefficients of \(x\) on both sides must be equal (and thus give us a contradiction). This leads us to set:

\[ 15 = 3 + a \]

Now, we solve for \(a\):

\[ a = 15 - 3 = 12 \]

Thus, the value of \([?]\) that would make the equation \(3 (5x + 6) = 3x + [?]x + 18\) have zero solutions is:

\[ \boxed{12} \]

Answer 2

alr bro good job next oneee!!

3 (4x + 4) = 2x + [?]x + [?]

Answer 3

To find the missing values that make the equation \(3 (4x + 4) = 2x + [?]x + [?]\) have zero solutions, we start by simplifying both sides of the equation.

Step 1: Simplify the left side.

\[ 3 (4x + 4) = 12x + 12 \]

Step 2: Rewrite the right side with the missing values represented as \(a\) and \(b\).

The right side is:

\[ 2x + ax + b = (2 + a)x + b \]

Step 3: Set the two sides equal to each other.

Now we have the equation:

\[ 12x + 12 = (2 + a)x + b \]

Step 4: Equate the coefficients and constants.

For the left side to equal the right side in a way that gives no solutions, the coefficients of \(x\) must be equal, but their constant parts must differ:

1. Equate coefficients of \(x\): \[ 12 = 2 + a \] Solving for \(a\): \[ a = 12 - 2 = 10 \]

2. Equate the constants: \[ 12 \neq b \] For zero solutions to occur, we can choose any value for \(b\) that is not 12.

Thus, we have:

• The missing coefficient of \(x\) is \(10\) (\([?] = 10\)).
• The constant term can be any value not equal to \(12\) (let's denote it as \(b\) and choose \(b = 0\) for simplicity).

Final result:

So the missing values are:

\[ \boxed{10 \text{ and } 0} \]

This gives us \(3(4x + 4) = 2x + 10x + 0\) or simply \(3(4x + 4) = 2x + 10x\) which will have zero solutions since both sides will have the same coefficient but different constant terms.

Answer 4

the 0 is wrong??

Answer 5

Apologies for any confusion! Let’s clarify the condition that leads to zero solutions in the equation.

Given:

\[ 3 (4x + 4) = 2x + [?]x + [?] \]

We already simplified the left side:

\[ 3 (4x + 4) = 12x + 12 \]

And we express the right side using the missing values \(a\) and \(b\):

\[ 2x + ax + b = (2 + a)x + b \]

Now, equate the two sides:

\[ 12x + 12 = (2 + a)x + b \]

Step 1: Compare the coefficients

1. For the \(x\) terms: \[ 12 = 2 + a \] Solving gives: \[ a = 12 - 2 = 10 \]

Step 2: Compare the constants

2. For the constant terms: \[ 12 = b \]

Conclusion: Condition for zero solutions

In order for the equation to have zero solutions, the coefficients of \(x\) must be equal, but the constant terms must differ. Therefore, we need:

• \(a = 10\) (as calculated)
• \(b\) must not equal \(12\).

Thus, we can say:

Final result:

1. The missing coefficient of \(x\) is \(\boxed{10}\)
2. The missing constant \(b\) must be any value except \(12\), such as \(\boxed{0}\), \(\boxed{1}\), or any number other than \(12\).

If you were looking for specific values for the constants, such as \(10\) (for the coefficient of \(x\)) and any number other than \(12\) (for the constant term), please choose a value for \(b\) that meets that condition. Thank you for your understanding, and please let me know if you have any further questions!

Answer 6

Determine the missing values that would make this equation have infinite amount of solutions.

Answer 7

To find the missing values that would make the equation \(3 (4x + 4) = 2x + [?]x + [?]\) have an infinite number of solutions, we need the left and right sides of the equation to be equivalent for any value of \(x\).

Step 1: Simplify the left side

We already simplified the left side:

\[ 3 (4x + 4) = 12x + 12 \]

Step 2: Rewrite the right side

The right side can be expressed as:

\[ 2x + ax + b = (2 + a)x + b \]

Step 3: Set the two sides equal to each other

For the two sides to be equivalent for all \(x\), both the coefficients of \(x\) must be equal and the constant terms must also be equal.

1. Set the coefficients of \(x\) equal:

\[ 12 = 2 + a \]

Solving for \(a\):

\[ a = 12 - 2 = 10 \]

2. Set the constant terms equal:

\[ 12 = b \]

Conclusion

To achieve an infinite number of solutions, we require:

• The coefficient of \(x\) is \(10\) (for \(a\)).
• The constant term \(b\) is \(12\).

Thus, the missing values are:

\[ \boxed{10 \text{ and } 12} \]

These values make both sides of the equation identical, leading to an infinite number of solutions for \(x\).