Question

A student’s work to solve an equation is shown.
​
1
8
(
40
x
+
16
)
=
9
x
−
7
(
2
x
−
1
)
−
5
5
x
+
2
=
9
x
−
14
x
+
7
−
5
5
x
+
2
=
5
x
+
2
2
=
2
8
1
​
(40x+16)=9x−7(2x−1)−5
5x+2=9x−14x+7−5
5x+2=5x+2
2=2
​
​Select the options that make the following statements true.
​
​ The student solved the equation
because the original equation has

correctly
incorrectly


no solution
one solution
infinitely many solutions

Answer 1

To analyze the student's work, let's break it down step by step:

1. The given equation is:
\[
\frac{1}{8}(40x + 16) = 9x - 7(2x - 1) - 5
\]

2. The student manipulates this to:
\[
5x + 2 = 9x - 14x + 7 - 5
\]

3. The student simplifies this to:
\[
5x + 2 = 5x + 2
\]

4. From the equation \(5x + 2 = 5x + 2\), we can see that this is a true statement. This means that for any value of \(x\), the equality holds true.

Now, to answer the statements:

- The student solved the equation **correctly** because the simplification led to a true statement.

- The original equation has **infinitely many solutions** because the equation simplifies to a statement that is always true (i.e., it does not exclude any value of \(x\)).

So the final selections would be:
- The student solved the equation **correctly**.
- The original equation has **infinitely many solutions**.