To show that the equation [tex]\( 4(x-6) + 10 = 7(x-2) - 3x \)[/tex] has an infinite number of solutions, we need to demonstrate that it simplifies to an identity, meaning both sides of the equation are equivalent for all values of [tex]\( x \)[/tex]. Here are two different approaches to prove this:
### Method 1: Expand and Simplify Both Sides
1. Expand and simplify the left-hand side (LHS):
[tex]\( 4(x-6) + 10 \)[/tex]
[tex]\[
= 4x - 24 + 10
\][/tex]
[tex]\[
= 4x - 14
\][/tex]
2. Expand and simplify the right-hand side (RHS):
[tex]\( 7(x-2) - 3x \)[/tex]
[tex]\[
= 7x - 14 - 3x
\][/tex]
[tex]\[
= 4x - 14
\][/tex]
3. Compare the simplified versions:
[tex]\[
4(x-6) + 10 = 4x - 14
\][/tex]
[tex]\[
7(x-2) - 3x = 4x - 14
\][/tex]
As we see from the steps above, the LHS and RHS simplify to the same expression [tex]\( 4x - 14 \)[/tex]. Since both sides of the equation are completely identical, the equation is an identity and holds true for any value of [tex]\( x \)[/tex]. Thus, it has an infinite number of solutions.
### Method 2: Equate the Coefficients
1. Rewrite the given equation:
[tex]\[
4(x-6) + 10 = 7(x-2) - 3x
\][/tex]
2. Expand both sides:
[tex]\[
4x - 24 + 10 = 7x - 14 - 3x
\][/tex]
[tex]\[
4x - 14 = 4x - 14
\][/tex]
3. Compare the coefficients and constant terms directly:
- The coefficient of [tex]\( x \)[/tex] on both sides is 4.
- The constant term on both sides is -14.
Since both sides are exactly the same, [tex]\( 4x - 14 = 4x - 14 \)[/tex], the equation holds for all [tex]\( x \)[/tex]. Hence, the equation [tex]\( 4(x-6) + 10 = 7(x-2) - 3x \)[/tex] is an identity and it has an infinite number of solutions.
By following either method, we can conclude that the equation is true for all values of [tex]\( x \)[/tex], and thus, there are an infinite number of solutions.
