What am I not seeing?

4x + y = 7
2x – 7y = 1

If I multiply the second equation by 2, I can stack them and subtract:

4x + y = 7
4x – 14y = 2

So, 15y = 5, —> y = 3

Then: 2x – 7(3) = 1 –> 2x – 21 = 3 —> 2x = 24 –> x = 12
But: 4x + 3 = 7 –> 4x = 4 –> x = 1

What am I not seeing? The answer should be x= 5/3.

The lengths of the sides of a rectangle are a and b…

The lengths of the sides of a rectangle are a and b, where a > b The sum of the lengths of the two shorter sides and one of the longer sides of the rectangle is 36. What value of a maximizes the area of the rectangle?

A .9
B. 12
C. 18
D. 24

The answer is C. I suspect there is an easier way to solve than completing the square and finding the vertex of the resulting quadratic function. What is your most direct, easy to understand solution to this calculator-allowed question?

The function f is defined by a polynomial…

x f(x)
____________
-5 0
-1 36
0 0
1 6
2 0

The function f is defined by a polynomial. Some values of x and f(x) are shown in the table above. Which of the following could define f?

A) (x – 5) (x + 2)
B) (x + 5)^2 (x – 2)^3
C) x^2 (x + 5) (x – 2)^2
D) x (x + 5) (x – 2)

Specifically, can you explain why C is correct but D cannot be?

In a circle with area of 120 to 124 sq inches…

In a circle with area of 120 to 124 sq inches, the area of the sector formed by an angle is between 20 and 21 sq inches. What is one possible integer value of the angle?

I came up with a low value of 60 (if area of circle is on the lowest end, 120 sq inches, and area of the sector is also on the low end, 20 inches). If both those areas are on the highest end, then I came up with 60 again. But answer is supposed to be 59 ≤ x ≤ 63. Does this make sense, and if so, can you explain it?

In the “Proving Grounds” Quiz 1 #5 explanation, you state: “…We know that, as a general rule, the remainder when a polynomial f(x) is divided by x – a is f(a).” Can you explain this further?

In the “Proving Grounds” Quiz 1 #5, this question is given:

p(x) = ax^2 + bx + c
q(x) = ax^2 + bx + d

The functions p(x) and q(x) are defined above, and a, b, c, and d are constants. If x – 3 is a factor of p(x) and d is 8 greater than c, what is the remainder when q(x) is divided by x – 3?

In your answer explanation, you state:
“…We know that, as a general rule, the remainder when a polynomial f(x) is divided by x – a is f(a).”

Can you explain this further?

In the system of equations above, a and b represent the distance, in meters, two marathon runners are…

a = 4800 – 6t
b = 5400 – 8t

In the system of equations above, a and b represent the distance, in meters, two marathon runners are from the finish line after running for four hours and t seconds. How far will runner a be from the finish line when runner b passes her?

A. 300 meters
B. 500 meters
C. 100 meters
D. 3000 meters

A rideshare app charges $2 per trip plus $0.4 per mile…

A rideshare app charges $2 per trip plus $0.4 per mile. A competitor charges $1 for the first 6 miles plus $0.5 per mile for every additional mile. For what length trip would the two services charge the same amount?

A. 10 miles
B. 18 miles
C. 20 miles
D. 40 miles

Can you craft an algebraic equation to solve this directly – or is plugging in the answers the way to go?