To determine whether the equation [tex]\(\frac{21}{x} = 7\)[/tex] has one solution, no solutions, or an infinite number of solutions, let's follow these steps:
1. Rewrite the Equation:
Start by rewriting the equation in a more manageable form:
[tex]\[
\frac{21}{x} = 7
\][/tex]
2. Isolate [tex]\(x\)[/tex]:
To solve for [tex]\(x\)[/tex], we can multiply both sides of the equation by [tex]\(x\)[/tex] to eliminate the fraction:
[tex]\[
21 = 7x
\][/tex]
3. Solve for [tex]\(x\)[/tex]:
Now divide both sides of the equation by 7 to isolate [tex]\(x\)[/tex]:
[tex]\[
x = \frac{21}{7}
\][/tex]
Simplifying the right-hand side gives:
[tex]\[
x = 3
\][/tex]
At this point, we have determined that there is exactly one solution, which is [tex]\(x = 3\)[/tex].
4. Verification:
Substitute [tex]\(x = 3\)[/tex] back into the original equation to verify our solution:
[tex]\[
\frac{21}{3} = 7
\][/tex]
Simplifying the left-hand side, we get:
[tex]\[
7 = 7
\][/tex]
This confirms that [tex]\( x = 3 \)[/tex] is indeed a solution.
Since we have found that [tex]\( x = 3 \)[/tex] is the solution and no other values of [tex]\( x \)[/tex] will satisfy the equation [tex]\(\frac{21}{x} = 7\)[/tex], we conclude that there is exactly one solution.
Thus, the equation [tex]\(\frac{21}{x} = 7\)[/tex] has one solution.
Therefore, a value of [tex]\(x\)[/tex] that makes the equation true is:
[tex]\[
x = 3
\][/tex]
There are no values of [tex]\(x\)[/tex] that make the equation false in the context of real numbers where [tex]\( x \neq 0 \)[/tex]. If another value [tex]\( x \)[/tex] is substituted into the original equation, it must be checked whether such values exist which make the equation true or false. For this solution context, we focus on the single solution shown.
