Question

steps to find the solution to an equation are shown in the table. move options to the table to show the property or reason for each step.

step	property or reason
$2x - 10 + 8x = 20 \to 10x - 10 = 20$
$10x - 10 = 20 \to 10x - 10 + 10 = 20 + 10$
$10x - 10 + 10 = 20 + 10 \to 10x = 30$
$10x = 30 \to \frac{1}{10}(10x) = \frac{1}{10}(30)$
$\frac{1}{10}(10x) = \frac{1}{10}(30) \to x = 3$

options:

• additive inverse property
• distributive property
• addition property of equality
• multiplication property of equality
• multiplicative inverse property
• combining like terms
• subtraction property of equality
• division property of equality

Explanation:

Step1: Identify property for first step

The first step is \(2(x - 5)+8x = 20
ightarrow2x - 10+8x = 20\). This is using the Distributive Property (\(a(b - c)=ab - ac\)), so we apply the Distributive Property here.

Step2: Identify property for second step

The second step is \(2x - 10+8x = 20
ightarrow10x - 10 = 20\). Here, we are combining like terms (\(2x+8x = 10x\)), so the reason is Combining Like Terms.

Step3: Identify property for third step

The third step is \(10x - 10 = 20
ightarrow10x - 10 + 10=20 + 10\). This is the Addition Property of Equality (adding the same number to both sides of an equation), so we use the Addition Property of Equality.

Step4: Identify property for fourth step

The fourth step is \(10x - 10 + 10=20 + 10
ightarrow10x = 30\). This is the Additive Inverse Property (since \(- 10+10 = 0\) and \(20 + 10-10=20\) in a way, but more precisely, adding the additive inverse to simplify), so we use the Additive Inverse Property.

Step5: Identify property for fifth step

The fifth step is \(10x = 30
ightarrow\frac{1}{10}(10x)=\frac{1}{10}(30)\). This is the Multiplication Property of Equality (multiplying both sides by the same non - zero number), so we use the Multiplication Property of Equality.

Step6: Identify property for sixth step

The sixth step is \(\frac{1}{10}(10x)=\frac{1}{10}(30)
ightarrow x = 3\). This is the Multiplicative Inverse Property (since \(\frac{1}{10}\times10 = 1\) and \(\frac{1}{10}\times30 = 3\)), so we use the Multiplicative Inverse Property.

Answer:

Step	Property or Reason

| \(2(x - 5)+8x = 20
ightarrow2x - 10+8x = 20\) | Distributive Property |
| \(2x - 10+8x = 20
ightarrow10x - 10 = 20\) | Combining Like Terms |
| \(10x - 10 = 20
ightarrow10x - 10 + 10=20 + 10\) | Addition Property of Equality |
| \(10x - 10 + 10=20 + 10
ightarrow10x = 30\) | Additive Inverse Property |
| \(10x = 30
ightarrow\frac{1}{10}(10x)=\frac{1}{10}(30)\) | Multiplication Property of Equality |
| \(\frac{1}{10}(10x)=\frac{1}{10}(30)
ightarrow x = 3\) | Multiplicative Inverse Property |