Question
2x - 5 = 8x + 7
4 solving steps
4 properties of multiplication used to solve it
4 solving steps
4 properties of multiplication used to solve it
Answers
GPT-4o mini
To solve the equation \(2x - 5 = 8x + 7\) in four steps and identify four properties of multiplication used, follow these steps:
### Solving Steps:
1. **Add 5 to both sides** (this is an application of the Addition Property of Equality):
\[
2x - 5 + 5 = 8x + 7 + 5 \implies 2x = 8x + 12
\]
2. **Subtract \(8x\) from both sides** (this is the Subtraction Property of Equality):
\[
2x - 8x = 8x - 8x + 12 \implies -6x = 12
\]
3. **Divide both sides by \(-6\)** (this utilizes the Division Property of Equality):
\[
\frac{-6x}{-6} = \frac{12}{-6} \implies x = -2
\]
4. **(Optional) Check the solution by substituting \(x = -2\) back into the original equation**:
\[
2(-2) - 5 = 8(-2) + 7 \implies -4 - 5 = -16 + 7 \implies -9 = -9 \quad (\text{True})
\]
### Properties of Multiplication Used:
1. **Addition Property of Equality**: Adding the same amount to both sides of an equation keeps the equation balanced (Step 1).
2. **Subtraction Property of Equality**: Subtracting the same amount from both sides of an equation maintains equality (Step 2).
3. **Division Property of Equality**: Dividing both sides of an equation by the same non-zero number gives an equivalent equation (Step 3).
4. **Distributive Property** (implied in operations): When rearranging terms or solving, it can be applied, especially when combining like terms.
If you specifically want to see multiplication properties, note that properties like the **Multiplicative Identity** (multiplying by 1) and **Multiplicative Inverse** (multiplying by \( \frac{1}{a} \) or dividing by a coefficient) are involved in maintaining the equality when we isolate \(x\). In this case, if we multiply or divide both sides by -1 at any point, it keeps the equality intact as well.
### Solving Steps:
1. **Add 5 to both sides** (this is an application of the Addition Property of Equality):
\[
2x - 5 + 5 = 8x + 7 + 5 \implies 2x = 8x + 12
\]
2. **Subtract \(8x\) from both sides** (this is the Subtraction Property of Equality):
\[
2x - 8x = 8x - 8x + 12 \implies -6x = 12
\]
3. **Divide both sides by \(-6\)** (this utilizes the Division Property of Equality):
\[
\frac{-6x}{-6} = \frac{12}{-6} \implies x = -2
\]
4. **(Optional) Check the solution by substituting \(x = -2\) back into the original equation**:
\[
2(-2) - 5 = 8(-2) + 7 \implies -4 - 5 = -16 + 7 \implies -9 = -9 \quad (\text{True})
\]
### Properties of Multiplication Used:
1. **Addition Property of Equality**: Adding the same amount to both sides of an equation keeps the equation balanced (Step 1).
2. **Subtraction Property of Equality**: Subtracting the same amount from both sides of an equation maintains equality (Step 2).
3. **Division Property of Equality**: Dividing both sides of an equation by the same non-zero number gives an equivalent equation (Step 3).
4. **Distributive Property** (implied in operations): When rearranging terms or solving, it can be applied, especially when combining like terms.
If you specifically want to see multiplication properties, note that properties like the **Multiplicative Identity** (multiplying by 1) and **Multiplicative Inverse** (multiplying by \( \frac{1}{a} \) or dividing by a coefficient) are involved in maintaining the equality when we isolate \(x\). In this case, if we multiply or divide both sides by -1 at any point, it keeps the equality intact as well.