Question
Complete the sentence based on the equation shown below.
14x + 14 =h (2x + 2)
This equation will have infinitely many solutions when h, = blank
because you get infinitely many solutions when you have blank number of x's on either side of the equation and blank
14x + 14 =h (2x + 2)
This equation will have infinitely many solutions when h, = blank
because you get infinitely many solutions when you have blank number of x's on either side of the equation and blank
Answers
Answered by
GPT-4o mini
To determine the value of \( h \) for which the equation \( 14x + 14 = h(2x + 2) \) will have infinitely many solutions, we first need to simplify the equation.
1. **Distribute \( h \) on the right side:**
\[
14x + 14 = h(2x + 2) = 2hx + 2h
\]
This gives us the equation:
\[
14x + 14 = 2hx + 2h
\]
2. **Rearrange the equation:**
\[
14x - 2hx = 2h - 14
\]
This can be rewritten as:
\[
(14 - 2h)x = 2h - 14
\]
For the equation to have infinitely many solutions, the coefficients of \( x \) must be equal, and the constant terms must be equal as well. Hence, we get the following conditions:
1. Coefficient of \( x \) must be zero:
\[
14 - 2h = 0 \implies 2h = 14 \implies h = 7
\]
2. The constant terms must also satisfy the equality:
If \( h = 7 \):
\[
2h - 14 = 0 \implies 2(7) - 14 = 0 \implies 14 - 14 = 0
\]
Thus, both conditions hold true with \( h = 7 \).
Now, we can fill in the blanks:
This equation will have infinitely many solutions when \( h = 7 \) because you get infinitely many solutions when you have **the same number of x's on either side of the equation** and **equal constant terms**.
1. **Distribute \( h \) on the right side:**
\[
14x + 14 = h(2x + 2) = 2hx + 2h
\]
This gives us the equation:
\[
14x + 14 = 2hx + 2h
\]
2. **Rearrange the equation:**
\[
14x - 2hx = 2h - 14
\]
This can be rewritten as:
\[
(14 - 2h)x = 2h - 14
\]
For the equation to have infinitely many solutions, the coefficients of \( x \) must be equal, and the constant terms must be equal as well. Hence, we get the following conditions:
1. Coefficient of \( x \) must be zero:
\[
14 - 2h = 0 \implies 2h = 14 \implies h = 7
\]
2. The constant terms must also satisfy the equality:
If \( h = 7 \):
\[
2h - 14 = 0 \implies 2(7) - 14 = 0 \implies 14 - 14 = 0
\]
Thus, both conditions hold true with \( h = 7 \).
Now, we can fill in the blanks:
This equation will have infinitely many solutions when \( h = 7 \) because you get infinitely many solutions when you have **the same number of x's on either side of the equation** and **equal constant terms**.