Question

Match the equation with its solution.

2i + 3 = 2i − 5
3(2i − 7) = 6i − 21
3i + 2i − 4 = 18i

options:

1). i = 8
2). infinite solutions
3). i = -1 over 3
4. no solution
5). i = 5
6). i = - 4 over 13

Answer 1

Look at the steps for solving the equation and choose a true statement.

3x − 9x + 1 = 2(−3x + 1) − 1

−6x + 1 = −6x + 2 − 1

−6x + 1 = −6x + 1

options:

A). There are infinitely many solutions to the equation.

B). The only solution to the equation is -6

C). There are no solutions to the equation.

D). The only solution to the equation is 1

E). There is no way to know if there is a solution to the equation.

Answer 2

a student solved the following equation using the following steps:

4(2 − 3x) = x − 2(2x + 1)

8 − 3x = x − 4x − 2

8 − 3x = −3x − 2

No solution.

based on the student's work, the equation was solved __________

The equation solved correctly would show that it has ____________
solution(s)

Answer 3

select the equation that has infinitely many solutions.

A). 3 − 3x = − 3(x − 1)

B). 4x + 1 = 4(x + 1)

C). 3x = 2x + 1

D). 3x + 5 = 3x − 5

Answer 4

A student solved the following equation using the following steps:

2(2x + 2) = 4(x + 4)

4x + 4 = 4x + 4

4 = 4

Infinitely many solutions

Based on the student's work, the equation was solved ___________

The equation solved correctly would show that it has ____________
solution(s).

Answer 5

The base of a triangle is four less than three times the height. The area of the triangle is 38 square centimeters. A = (b × h) over 2

An expression for the base of the triangle: _______________

Equation for the area of the triangle, substituting in for the base: __________

The first step in solving for h: __________________

options:
4 - 3h
4h - 3
divide both sides by h
3h - 4
multiple both sides by 2
3h - 4 = 1 over 2 (38 x h)
38 = (3h - 4 x h) over

Answer 6

Solve the equation for y: 3x+5y=60

options:
1. y = 60 − 3x

2. y = 60 + 3x over 5

3. y=−3 over 5 x + 12

4. y = 57x over 5

Answer 7

Solve the following:

ax − 44 < 15

ax + 15 = 44

option:
x = 29a
x = 29 over a
x = 59 over a
x < 59 over a
x < 29a
x < 59a
x = 59a
x < 29 over a

Answer 8

Let's solve each equation one by one and match them to the appropriate solutions.

1. **Equation**: \( 2i + 3 = 2i - 5 \)

Subtract \( 2i \) from both sides:
\[ 3 = -5 \]

This is a false statement, indicating that there are no values of \( i \) that satisfy the equation.

- **Match**: 4). no solution


2. **Equation**: \( 3(2i - 7) = 6i - 21 \)

Distributing on the left side:
\[ 6i - 21 = 6i - 21 \]

The left side equals the right side for any value of \( i \). Therefore, there are infinite solutions.

- **Match**: 2). infinite solutions


3. **Equation**: \( 3i + 2i - 4 = 18i \)

Combine like terms:
\[ 5i - 4 = 18i \]

Subtract \( 5i \) from both sides:
\[ -4 = 13i \]

Now solve for \( i \):
\[ i = -\frac{4}{13} \]

- **Match**: 6). i = -4 over 13


Now we have matched the equations with their solutions:

1. **4** (no solution)
2. **2** (infinite solutions)
3. **6** (i = -4 over 13)

Answer 9

Let's analyze the steps provided in solving the equation:

1. **Original Equation**:
\[ 3x - 9x + 1 = 2(-3x + 1) - 1 \]

2. **First Step**:
\[ -6x + 1 = -6x + 2 - 1 \]
This simplifies to:
\[ -6x + 1 = -6x + 1 \]

This shows that both sides of the equation are identical at this point. This implies that the equation is true for all values of \( x \).

Therefore, there are infinitely many solutions to the equation because both sides are equal regardless of the value of \( x \).

- **True Statement**:
A). There are infinitely many solutions to the equation.

Answer 10

Let's analyze the student's work step by step:

1. **Original Equation**:
\[ 4(2 - 3x) = x - 2(2x + 1) \]

2. **First Step**:
Distributing on both sides:
\[ 8 - 12x = x - 4x - 2 \]
Simplifying the right side gives:
\[ 8 - 12x = -3x - 2 \]

3. **Second Step**:
Now we have:
\[ 8 - 12x = -3x - 2 \]

4. **Rearranging**:
Add \( 12x \) to both sides:
\[ 8 = 9x - 2 \]

Now, add \( 2 \) to both sides:
\[ 10 = 9x \]

Now, divide by \( 9 \):
\[ x = \frac{10}{9} \]

This means the solution is \( x = \frac{10}{9} \), indicating that there is one valid solution.

The student's work incorrectly concluded "No solution" before solving properly.

**Based on the student's work**,
The equation was solved **incorrectly**.

The equation solved correctly would show that it has **1 solution**.

Answer 11

To determine which equation has infinitely many solutions, we should analyze each equation step by step.

### A) \( 3 - 3x = -3(x - 1) \)

Distributing on the right side:
\[ 3 - 3x = -3x + 3 \]

When we simplify:
\[ 3 - 3x = -3x + 3 \]

This is true for all \( x \) because both sides are equal.

### B) \( 4x + 1 = 4(x + 1) \)

Distributing on the right side:
\[ 4x + 1 = 4x + 4 \]

Subtracting \( 4x \) from both sides gives:
\[ 1 = 4 \]

This is a false statement, indicating no solutions.

### C) \( 3x = 2x + 1 \)

Subtracting \( 2x \) from both sides gives:
\[ x = 1 \]

This has a single solution.

### D) \( 3x + 5 = 3x - 5 \)

Subtracting \( 3x \) from both sides gives:
\[ 5 = -5 \]

This is a false statement, indicating no solutions.

### Conclusion

Among the equations:

- **A)** has infinitely many solutions.
- **B)** has no solutions.
- **C)** has one solution.
- **D)** has no solutions.

**The equation that has infinitely many solutions is:**

**A) \( 3 - 3x = -3(x - 1) \)**

Answer 12

Let's analyze the student's work step by step:

1. **Original Equation**:
\[ 2(2x + 2) = 4(x + 4) \]

2. **Distributing**:
The left side becomes:
\[ 4x + 4 \]
The right side becomes:
\[ 4x + 16 \]

So, the equation simplifies to:
\[ 4x + 4 = 4x + 16 \]

3. **Next Step**:
When the student wrote:
\[ 4x + 4 = 4x + 4 \]
This is incorrect because the right side should have stayed as \( 4x + 16 \).

4. **Final Comparison**:
The student mistakenly equated equal expressions, leading to an erroneous statement \( 4 = 4 \), which simplifies correctly but doesn't address the original equation properly.

Since the left side \( 4x + 4 \) does not equal \( 4x + 16 \), we ultimately have the statement \( 4 \neq 16 \), indicating that there are actually **no solutions** to the original equation.

### Conclusion

- **Based on the student's work**, the equation was solved **incorrectly**.

- **The equation solved correctly would show that it has** **no solution**.

Answer 13

Let's break down the problem step by step.

### Step 1: Expression for the base of the triangle

Given that the base of the triangle (\(b\)) is four less than three times the height (\(h\)), we can express the base as:

\[
b = 3h - 4
\]

So, the expression for the base of the triangle is:
**Expression for the base of the triangle:** **3h - 4**

---

### Step 2: Equation for the area of the triangle

The formula for the area of a triangle is:
\[
A = \frac{b \times h}{2}
\]

Substituting the expression for the base (\(3h - 4\)) into the area formula and setting the area \(A\) to 38 square centimeters:

\[
38 = \frac{(3h - 4) \times h}{2}
\]

So, the equation for the area of the triangle is:
**Equation for the area of the triangle, substituting in for the base:** **38 = (3h - 4) \cdot h / 2**

---

### Step 3: The first step in solving for \(h\)

To eliminate the fraction in the equation, the first step in solving for \(h\) is to multiply both sides by 2:

\[
2 \cdot 38 = (3h - 4) \cdot h
\]

So, the first step in solving for \(h\) is:
**The first step in solving for h:** **multiply both sides by 2**

---

To summarize:
1. Expression for the base: **3h - 4**
2. Equation for the area: **38 = (3h - 4) \cdot h / 2**
3. First step in solving for \(h\): **multiply both sides by 2**

Answer 14

To solve the equation \( 3x + 5y = 60 \) for \( y \), we need to isolate \( y \) on one side of the equation. Here are the steps:

1. **Start with the original equation**:
\[
3x + 5y = 60
\]

2. **Subtract \( 3x \) from both sides**:
\[
5y = 60 - 3x
\]

3. **Now, divide every term by 5 to solve for \( y \)**:
\[
y = \frac{60 - 3x}{5}
\]

This can be separated into:
\[
y = \frac{60}{5} - \frac{3x}{5}
\]

Which simplifies to:
\[
y = 12 - \frac{3}{5}x
\]

Thus, the rearranged equation is:
\[
y = -\frac{3}{5}x + 12
\]

### Conclusion

The correct option is:
**3. \( y = -\frac{3}{5}x + 12 \)**

Answer 15

Let's solve the given inequalities and equations step by step.

### Step 1: Solve the inequality \( ax - 44 < 15 \)

First, we isolate \( ax \) on one side:

1. Add 44 to both sides:
\[
ax < 15 + 44
\]
\[
ax < 59
\]

2. **Now divide both sides by \( a \)** (assuming \( a \neq 0 \)):
- If \( a > 0 \):
\[
x < \frac{59}{a}
\]

- If \( a < 0 \):
\[
x > \frac{59}{a} \quad \text{(note that the direction of the inequality changes)}
\]

For this problem, we'll assume \( a > 0\) to focus on the common scenario:

\[
x < \frac{59}{a}
\]

### Step 2: Solve the equation \( ax + 15 = 44 \)

1. Subtract 15 from both sides:
\[
ax = 44 - 15
\]
\[
ax = 29
\]

2. Now divide both sides by \( a \) (again assuming \( a \neq 0 \)):
\[
x = \frac{29}{a}
\]

### Summary of Results

- From the inequality \( ax - 44 < 15 \):
- **If \( a > 0 \)**: \( x < \frac{59}{a} \)
- The solution for the equation \( ax + 15 = 44 \):
- \( x = \frac{29}{a} \)

### Conclusion
- The correct option for the inequality is:
**\( x < \frac{59}{a} \)**

Thus, the solution to the inequality is represented by **\( x < \frac{59}{a} \)**.