Question

1. from the given bank of statements, determine the steps needed to solve the following equations in the correct order. write the correct letter sequence on the blank provided.

a. $5 - x = 12$
bank
a) add 5 to both sides.
b) subtract 5 from both sides.
c) multiply both sides by 5.
d) divide both sides by 5.
e) add 1 to both sides.
f) subtract 1 from both sides.
g) divide both sides by 1.
h) divide both sides by -1.

b. $14 = \frac{4}{5}x - 2$
bank:
a) multiply both sides by 4.
b) divide both sides by 4.
c) multiply both sides by 5.
d) divide both sides by 5.
e) add $\frac{4}{5}$ to both sides.
f) subtract $\frac{4}{5}$ from both sides.
g) multiply both sides by $\frac{4}{5}$.
h) divide both sides by $\frac{4}{5}$.
i) add 2 to both sides.
j) subtract 2 from both sides.

c. $\frac{2x + 4}{3} = 9$
bank
a) add 2 to both sides.
b) subtract 2 from both sides.
c) multiply both sides by 2.
d) divide both sides by 2.
e) add 4 to both sides.
f) subtract 4 from both sides.
g) add 3 to both sides.
h) subtract 3 from both sides.
i) multiply both sides by 3.
j) divide both sides by 3.

Explanation:

Part a: Solving \( 5 - x = 12 \)

Step 1: Subtract 5 from both sides

To isolate the term with \( x \), we first subtract 5 from both sides of the equation \( 5 - x = 12 \).
\( 5 - x - 5 = 12 - 5 \)
Simplifying, we get \( -x = 7 \).

Step 2: Divide both sides by -1

To solve for \( x \), we divide both sides of the equation \( -x = 7 \) by -1.
\( \frac{-x}{-1} = \frac{7}{-1} \)
Simplifying, we get \( x = -7 \).

The correct sequence of steps from the bank is: B (Subtract 5 from both sides) and then H (Divide both sides by -1).

Part b: Solving \( 14 = \frac{4}{5}x - 2 \)

Step 1: Add 2 to both sides

To isolate the term with \( x \), we first add 2 to both sides of the equation \( 14 = \frac{4}{5}x - 2 \).
\( 14 + 2 = \frac{4}{5}x - 2 + 2 \)
Simplifying, we get \( 16 = \frac{4}{5}x \).

Step 2: Multiply both sides by 5

Next, we multiply both sides of the equation \( 16 = \frac{4}{5}x \) by 5 to eliminate the fraction.
\( 16 \times 5 = \frac{4}{5}x \times 5 \)
Simplifying, we get \( 80 = 4x \).

Step 3: Divide both sides by 4

Finally, we divide both sides of the equation \( 80 = 4x \) by 4 to solve for \( x \).
\( \frac{80}{4} = \frac{4x}{4} \)
Simplifying, we get \( x = 20 \).

The correct sequence of steps from the bank is: J (Add 2 to both sides) and then G (Multiply both sides by \( \frac{5}{4} \)) or equivalently, adding 2 (J), then multiplying by 5 (C), then dividing by 4 (B). Wait, looking at the bank options for part b:

Bank options for part b:
A) Multiply both sides by 4.
B) Divide both sides by 4.
C) Multiply both sides by 5.
D) Divide both sides by 5.
E) Add \( \frac{4}{5} \) to both sides.
F) Subtract \( \frac{4}{5} \) from both sides.
G) Multiply both sides by \( \frac{5}{4} \).
H) Divide both sides by \( \frac{4}{5} \).
I) Add 2 to both sides.
J) Subtract 2 from both sides.

So the correct steps are: I (Add 2 to both sides) to get \( 16 = \frac{4}{5}x \), then G (Multiply both sides by \( \frac{5}{4} \)) to solve for \( x \).

Part c: Solving \( \frac{2x + 4}{3} = 9 \)

Step 1: Multiply both sides by 3

To eliminate the denominator, we multiply both sides of the equation \( \frac{2x + 4}{3} = 9 \) by 3.
\( \frac{2x + 4}{3} \times 3 = 9 \times 3 \)
Simplifying, we get \( 2x + 4 = 27 \).

Step 2: Subtract 4 from both sides

Next, we subtract 4 from both sides of the equation \( 2x + 4 = 27 \) to isolate the term with \( x \).
\( 2x + 4 - 4 = 27 - 4 \)
Simplifying, we get \( 2x = 23 \).

Step 3: Divide both sides by 2

Finally, we divide both sides of the equation \( 2x = 23 \) by 2 to solve for \( x \).
\( \frac{2x}{2} = \frac{23}{2} \)
Simplifying, we get \( x = \frac{23}{2} = 11.5 \).

The correct sequence of steps from the bank is: D (Multiply both sides by 3), then F (Subtract 4 from both sides), then D (Divide both sides by 2). Wait, looking at the bank options for part c:

Bank options for part c:
A) Add 2 to both sides.
B) Subtract 2 from both sides.
C) Multiply both sides by 2.
D) Divide both sides by 2.
E) Add 4 to both sides.
F) Subtract 4 from both sides.
G) Add 3 to both sides.
H) Subtract 3 from both sides.
I) Multiply both sides by 3.
J) Divide both sides by 3.

So the correct steps are: I (Multiply both sides by 3) to get \( 2x + 4 = 27 \), then F (Subtract 4 from both sides) to get \( 2x = 23 \), then D (Divide both sides by 2) to solve for \( x \).

Answer:

s:
a. The correct sequence is B (Subtract 5 from both sides) and H (Divide both sides by -1). So the letter sequence is B, H.
b. The correct sequence is I (Add 2 to both sides) and G (Multiply both sides by \( \frac{5}{4} \)). So the letter sequence is I, G.
c. The correct sequence is I (Multiply both sides by 3), F (Subtract 4 from both sides), and D (Divide both sides by 2). So the letter sequence is I, F, D.