Complete the sentence based on the equation shown below.

Question

-8 (x + 3) = h ( -x + 10)

This equation will have zero solutions when h = blank because you get zero solutions when you have blank number of x's on either side of the equation and blank equation and blank

Answer 1

To determine the value of \( h \) for which the equation \(-8 (x + 3) = h ( -x + 10)\) has zero solutions, we can analyze the equation step by step.

1. Distribute the terms:

   • Left side: \(-8 (x + 3) = -8x - 24\)
   • Right side: \(h ( -x + 10) = -hx + 10h\)

2. Rewrite the equation: \[ -8x - 24 = -hx + 10h \]

3. Rearranging: Rearranging the equation gives: \[ -8x + hx = 10h + 24 \] \[ (h - 8)x = 10h + 24 \]

For the equation to have zero solutions, we want the coefficient of \(x\) to be zero. This means:

• The term \((h - 8)\) should equal zero: \[ h - 8 = 0 \implies h = 8 \]

At the same time, the constant term on the right-hand side must not equal zero:

• The equation then is: \[ 10h + 24 \neq 0 \] Substituting \(h = 8\): \[ 10(8) + 24 = 80 + 24 = 104 \neq 0 \]

Thus, the complete sentence would be:

This equation will have zero solutions when \(h = 8\) because you get zero solutions when you have one number of \(x\) on either side of the equation and a non-zero equation for the constants.

Answer 2

the second blank and the third on are wrong

Options for the second blank- the same or different
Options for the third blank- the same constant, the different constant, or constants can be the same or different

Answer 3

Thanks for the clarification! Let’s fill in the blanks appropriately based on your options:

This equation will have zero solutions when \(h = 8\) because you get zero solutions when you have the same number of \(x\)'s on either side of the equation and different constants.

Answer 4

do the same thing for this problem
12x - 12 = h (4x + 8)

Answer 5

To determine the value of \( h \) for which the equation \( 12x - 12 = h (4x + 8) \) has zero solutions, we can analyze the equation step by step.

1. Distribute the terms:

   • Right side: \( h(4x + 8) = 4hx + 8h \)

2. Rewrite the equation: \[ 12x - 12 = 4hx + 8h \]

3. Rearranging: Rearranging the equation gives: \[ 12x - 4hx = 8h + 12 \] \[ (12 - 4h)x = 8h + 12 \]

For the equation to have zero solutions, we want the coefficient of \( x \) to be zero. This means:

• The term \( (12 - 4h) \) should equal zero: \[ 12 - 4h = 0 \implies 4h = 12 \implies h = 3 \]

At the same time, the constant term on the right-hand side must not equal zero:

• The equation then is: \[ 8h + 12 \neq 0 \] Substituting \( h = 3 \): \[ 8(3) + 12 = 24 + 12 = 36 \neq 0 \]

Thus, the complete sentence would be:

This equation will have zero solutions when \(h = 3\) because you get zero solutions when you have the same number of \(x\)'s on either side of the equation and different constants.

Answer 6

Last one
h (2x + 6) = 14 (x + 3)

Answer 7

To determine the value of \( h \) for which the equation \( h(2x + 6) = 14(x + 3) \) has zero solutions, let's analyze the equation step by step.

1. Distribute the terms:

   • Right side: \( 14(x + 3) = 14x + 42 \)

2. Rewrite the equation: \[ h(2x + 6) = 14x + 42 \]

3. Rearranging: This can be written as: \[ 2hx + 6h = 14x + 42 \]

4. Group the terms: Rearranging gives: \[ 2hx - 14x = 42 - 6h \] \[ (2h - 14)x = 42 - 6h \]

For the equation to have zero solutions, we want the coefficient of \( x \) to be zero, which means:

• The term \( (2h - 14) \) should equal zero: \[ 2h - 14 = 0 \implies 2h = 14 \implies h = 7 \]

At the same time, the constant term on the right-hand side must not equal zero:

• The equation then is: \[ 42 - 6h \neq 0 \] Substituting \( h = 7 \): \[ 42 - 6(7) = 42 - 42 = 0 \]

This means that \( h = 7 \) does not satisfy the conditions for zero solutions since it leads to both sides having \( x \) terms cancelled out and the constants becoming equal (which gives infinite solutions instead). Thus, there is no \( h \) where this condition holds true.

However, we can choose \( h \) such that the equation becomes undefined.

Final Conclusion:

Therefore, since there is no value of \( h \) that gives exactly zero solutions while satisfying different constant conditions, we can clarify that:

This equation will not have solutions when \( h = 7 \) because you get zero solutions when you have the same number of \( x \)'s on either side of the equation and the same constant. In this case, the constants become equal to each other when \( h = 7 \).

If the equation is meant to have no solutions instead of infinite solutions, we can consider values where:

1. \( 2h - 14 \neq 0 \),
2. and \( 42 - 6h = 0\) (which leads to the above condition).

If you'd like to finalize on a solution that avoids \( h \) being 7 entirely, please let me know!