To determine how many solutions the given equation
[tex]\[ 3x + 7 + 2x = 5x + 7 \][/tex]
has, let's proceed with the following steps:
1. Combine like terms on the left-hand side of the equation. Notice that [tex]\(3x\)[/tex] and [tex]\(2x\)[/tex] are like terms:
[tex]\[ 3x + 2x + 7 = 5x + 7 \][/tex]
Adding [tex]\(3x\)[/tex] and [tex]\(2x\)[/tex] gives:
[tex]\[ 5x + 7 = 5x + 7 \][/tex]
2. Simplify the equation by subtracting [tex]\(5x\)[/tex] from both sides:
[tex]\[ 5x + 7 - 5x = 5x + 7 - 5x \][/tex]
This simplifies to:
[tex]\[ 7 = 7 \][/tex]
3. This result is an identity. An identity means that both sides of the equation are equal for all values of [tex]\(x\)[/tex]. In this particular case, the equation does not depend on [tex]\(x\)[/tex] anymore and is just a true statement (7 equals 7).
Because the simplified equation is always true regardless of the value of [tex]\(x\)[/tex], it implies that there are infinitely many solutions to the original equation.
To summarize:
- The original equation simplifies to an identity [tex]\(7 = 7\)[/tex].
- This means that the equation holds true for any value of [tex]\(x\)[/tex].
Hence, there are infinitely many solutions.
