Andrea is 6 years younger than Tony. In two years, Tony will be three times as old as Andrea. Let a be Tony’s current age. Which of these linear equations can be used to find Andrea’s and Tony’s current ages?

Question

options:

A). a = 3 (a + 2)

B). a + 2 = 3 (a + 6 + 2)

C). 3(a + 2) = a − 6 + 2

D). a + 2 = 3(a − 6 + 2)

Answer 1

Let \( a \) be Tony's current age. Since Andrea is 6 years younger than Tony, her current age can be expressed as:

\[ a - 6 \]

In two years, Tony's age will be:

\[ a + 2 \]

And Andrea's age in two years will be:

\[ (a - 6) + 2 = a - 4 \]

According to the problem, in two years, Tony will be three times as old as Andrea:

\[ a + 2 = 3(a - 4) \]

Now we will examine each of the provided options to see which one matches this equation.

Option A: \[ a = 3(a + 2) \] This does not match our equation.

Option B: \[ a + 2 = 3(a + 6 + 2) \] This simplifies to \( a + 2 = 3(a + 8) \), which is not consistent with our equation.

Option C: \[ 3(a + 2) = a - 6 + 2 \] This simplifies to \( 3(a + 2) = a - 4 \), which is not correct.

Option D: \[ a + 2 = 3(a - 6 + 2) \] This simplifies to \( a + 2 = 3(a - 4) \), which is what we derived.

Therefore, the correct option is:

D). \( a + 2 = 3(a - 6 + 2) \)

Answer 2

Five less than three-fourths of x is seven. If x equals 16, what is the last property of equality you would use to justify the solution to this problem?

Answer 3

To solve the equation "Five less than three-fourths of \( x \) is seven," we can express this mathematically as:

\[ \frac{3}{4}x - 5 = 7 \]

Now, let's solve for \( x \):

1. Add 5 to both sides:

\[ \frac{3}{4}x - 5 + 5 = 7 + 5 \]

This simplifies to:

\[ \frac{3}{4}x = 12 \]

2. Now, multiply both sides by \(\frac{4}{3}\) to isolate \( x \):

\[ x = 12 \times \frac{4}{3} \]

Calculating the right side gives:

\[ x = 16 \]

So, we find the value of \( x \) to be 16.

If we want to reason about the last property of equality used in this solution, it happens at the final steps when we isolated \( x \). The last operation we performed was multiplying both sides of the equation by \(\frac{4}{3}\).

The property of equality being applied here is the Multiplication Property of Equality, which states that if you multiply both sides of an equation by the same nonzero number, the two sides remain equal.

Therefore, the last property of equality you would use to justify the solution is the Multiplication Property of Equality.