Question
Suzie solved the equation 9x−5=14(16x+60) using the following steps
Distribute 1/4 to get 9x−5=4x+60
Use subtraction property of equality to move 4x to get 5x−5=60
Use addition property of equality to move 5 and get 5x=65
Use division property of equality to get x=13
Determine if Suzie solved correctly - if not, identify which step was the mistake.
Distribute 1/4 to get 9x−5=4x+60
Use subtraction property of equality to move 4x to get 5x−5=60
Use addition property of equality to move 5 and get 5x=65
Use division property of equality to get x=13
Determine if Suzie solved correctly - if not, identify which step was the mistake.
Answers
GPT-4o mini
Let's carefully analyze the steps Suzie took to solve the equation \(9x - 5 = 14(16x + 60)\).
1. **Start with the original equation**:
\[
9x - 5 = 14(16x + 60)
\]
2. **Distributing \(14\) on the right-hand side**:
Suzie wrote \(9x − 5 = 4x + 60\). However, we need to distribute \(14\) through \((16x + 60)\):
\[
14 \times 16x = 224x \quad \text{and} \quad 14 \times 60 = 840
\]
Therefore:
\[
14(16x + 60) = 224x + 840
\]
The correct equation is:
\[
9x - 5 = 224x + 840
\]
3. **Subsequent Steps**:
Suzie then tried to move \(4x\) from the right-hand side, but she used the wrong equation. Let's continue solving correctly from:
\[
9x - 5 = 224x + 840
\]
4. **Using subtraction property of equality to isolate \(x\)**:
Subtract \(224x\) from both sides:
\[
9x - 224x - 5 = 840
\]
This simplifies to:
\[
-215x - 5 = 840
\]
5. **Using addition property of equality**:
Add \(5\) to both sides:
\[
-215x = 845
\]
6. **Using division property of equality**:
Finally, divide by \(-215\):
\[
x = \frac{845}{-215}
\]
This could be simplified depending on the numbers involved, but it will yield a different result than what Suzie found.
**Conclusion**:
Suzie's mistake occurred in the first step, where she incorrectly distributed \(14\) through \((16x + 60)\). Instead of getting \(9x - 5 = 4x + 60\), the correct equation after distribution should have been \(9x - 5 = 224x + 840\). Therefore, all subsequent steps were based on incorrect information, leading to the wrong answer.
1. **Start with the original equation**:
\[
9x - 5 = 14(16x + 60)
\]
2. **Distributing \(14\) on the right-hand side**:
Suzie wrote \(9x − 5 = 4x + 60\). However, we need to distribute \(14\) through \((16x + 60)\):
\[
14 \times 16x = 224x \quad \text{and} \quad 14 \times 60 = 840
\]
Therefore:
\[
14(16x + 60) = 224x + 840
\]
The correct equation is:
\[
9x - 5 = 224x + 840
\]
3. **Subsequent Steps**:
Suzie then tried to move \(4x\) from the right-hand side, but she used the wrong equation. Let's continue solving correctly from:
\[
9x - 5 = 224x + 840
\]
4. **Using subtraction property of equality to isolate \(x\)**:
Subtract \(224x\) from both sides:
\[
9x - 224x - 5 = 840
\]
This simplifies to:
\[
-215x - 5 = 840
\]
5. **Using addition property of equality**:
Add \(5\) to both sides:
\[
-215x = 845
\]
6. **Using division property of equality**:
Finally, divide by \(-215\):
\[
x = \frac{845}{-215}
\]
This could be simplified depending on the numbers involved, but it will yield a different result than what Suzie found.
**Conclusion**:
Suzie's mistake occurred in the first step, where she incorrectly distributed \(14\) through \((16x + 60)\). Instead of getting \(9x - 5 = 4x + 60\), the correct equation after distribution should have been \(9x - 5 = 224x + 840\). Therefore, all subsequent steps were based on incorrect information, leading to the wrong answer.