Question

The first step to solve for y in y/3
= 6 + x is
Answer
Consider the sequence of steps to solve the equation: 3(x − 4) + 5x = 9x − 36

Given ⇒ 3(x − 4) + 5x = 9x − 36
Step 1 ⇒ 3x − 12 + 5x = 9x − 36
Step 2 ⇒ 3x + 5x − 12 = 9x − 36
Step 3 ⇒ 8x − 12 = 9x − 36
Step 4 ⇒ 8x − 8x − 12 = 9x − 8x − 36
Step 5 ⇒ 0 − 12 = x − 36
Step 6 ⇒ −12 = x − 36
Step 7 ⇒ −12 + 36 = x − 36 + 36
Step 8 ⇒ 24 = x + 0
Step 9 ⇒ 24 = x

Which property yields Step 7?
Answer
Consider the sequence of steps to solve the equation:
5(x - 3) = 7x/2


Step 1 ⇒ 10(x - 3) = 7x
Step 2 ⇒ 10x - 30 = 7x
Step 3 ⇒ 3x - 30 = 0
Step 4 ⇒ 3x = 30
Step 5 ⇒ x = 10

Identify the property of equality which gets us from Step 3 to Step 4.
Answer
Step 1: 4x + 5 < 6x + 1 (Given)
Step 2: −2x + 5 < 1 (Subtraction)
Step 3: −2x < 6 (Addition)
Step 4: x > −3 (Division)

Mia is trying to find her mistake in the problem shown. In which step did she first make a mistake?
Answer
4x + 5x + 12 = -6 (given)
9x + 12 = -6 (simplify)
9x = -18 (subtraction)
x = -2 (addition)

Which reason is incorrect?
Answer
Solving the equation 6x + 2(x − 1) = 30

Given: 6x + 2(x − 1) = 30
Step 1: 6x + 2x − 2 = 30
Step 2: 8x − 2 = 30
Step 3: 8x − 2 + 2 = 30 + 2
Step 4: 8x/8 = 32/8
Step 5: x = 4

Identify the property that justifies Step 1 when solving the given equation.
Answer
Identify the property that justifies Step 2 when solving the given equation.
Answer
A Reflexive Property
B Substitution Property
C Symmetric Property
D Multiplication Property
Answered by GPT 3.5
To solve for y in the equation y/3 = 6 + x, we can start by multiplying both sides of the equation by 3 to get rid of the fraction.

This gives us y = 3*(6 + x).

Next, we distribute the 3 to each term inside the parentheses:

y = 3*6 + 3*x.

Simplifying further, we have:

y = 18 + 3x.

Therefore, the first step to solve for y is to rewrite the equation as y = 18 + 3x.
Answered by GPT 3.5
Step 7 is an example of the addition property of equality. This property states that if you add the same value to both sides of an equation, the equation remains true. In this case, we added 36 to both sides of the equation to get rid of the -36 on the right side.
Answered by GPT 3.5
The property of equality that gets us from Step 3 to Step 4 is the addition property of equality. This property states that if you add or subtract the same value to both sides of an equation, the equation remains true. In this case, we added 30 to both sides of the equation to isolate the variable x on one side.
Answered by GPT 3.5
Mia made her first mistake in Step 3: −2x < 6.

To get from Step 2 to Step 3, Mia subtracted 5 from both sides of the inequality. However, she made an error in the sign of the inequality. When subtracting a positive number, the inequality sign should not change. So, the correct step should be −2x + 5 > 1.
Answered by GPT 3.5
The incorrect reason is "9x = -18 (subtraction)".

To get from the equation 9x + 12 = -6 to 9x = -18, you need to subtract 12 from both sides, not subtract 6.

Therefore, the correct reasoning should be:
9x + 12 - 12 = -6 - 12
9x = -18
Answered by GPT 3.5
The property that justifies Step 1 when solving the given equation is the distributive property.

The distributive property states that for any real numbers a, b, and c, a(b + c) = ab + ac. Applying this property to the equation 6x + 2(x - 1) = 30, we distribute the 2 to both terms inside the parentheses:

6x + 2x - 2 = 30

This results in multiplying 2 by both x and -1, which is why we have 2x - 2 on the left side of the equation.
Answered by GPT 3.5
The property that justifies Step 2 when solving the given equation is the combination of like terms.

In Step 2, we combine the like terms 6x and 2x to simplify the expression. By adding the coefficients of x, we get 8x. The equation becomes:

8x - 2 = 30
Answered by GPT 3.5
The property that justifies Step 2 when solving the given equation is the Substitution Property.

In Step 2, we substitute the value of 2(x - 1) with 2x into the equation. This allows us to simplify the expression by replacing the original term with an equivalent term. The equation becomes:

6x + 2x - 2 = 30