Question

Why is the system of equations 2x+7y=14\:and\:4x+5y=10 equivalent to 4x+14y=28\:and\:8x+10y=20
A. the intercepts of 2x+7y=14 and 4x+14y=28 are (0, 2) and (7, 0)
B. Both equations, 2x+7y=14 and 4x+5y=10, can be multiplied by 1/2 to produce the equivalent system of equations.
C. The intercepts of both 4x+5y=10 and 8x+10y=20 are (0, 2.5) and (2,0)
D. All equations share the point (0, 2)

Answer 1

To determine why the system of equations \(2x + 7y = 14\) and \(4x + 5y = 10\) is equivalent to the system \(4x + 14y = 28\) and \(8x + 10y = 20\), let's analyze each option.

1. **Option A: The intercepts of \(2x + 7y = 14\) and \(4x + 14y = 28\) are \((0, 2)\) and \((7, 0)\)**.

- Setting \(x = 0\) in \(2x + 7y = 14\):
\[
2(0) + 7y = 14 \implies 7y = 14 \implies y = 2 \implies (0, 2)
\]
- Setting \(y = 0\) in \(2x + 7y = 14\):
\[
2x + 7(0) = 14 \implies 2x = 14 \implies x = 7 \implies (7, 0)
\]
- For \(4x + 14y = 28\), these calculations lead to the same intercepts, but this does not imply equivalence to the other system.

2. **Option B: Both equations, \(2x + 7y = 14\) and \(4x + 5y = 10\), can be multiplied by \(\frac{1}{2}\) to produce the equivalent system of equations.**

- The first equation \(2x + 7y = 14\) multiplied by \(2\) gives \(4x + 14y = 28\).
- The second equation \(4x + 5y = 10\) multiplied by \(2\) gives \(8x + 10y = 20\).
- Therefore, this transformation holds, making this choice valid for demonstrating equivalence.

3. **Option C: The intercepts of both \(4x + 5y = 10\) and \(8x + 10y = 20\) are \((0, 2.5)\) and \((2,0)\)**.

- The \(y\)-intercept of \(4x + 5y = 10\) occurs when \(x = 0\):
\[
4(0) + 5y = 10 \implies 5y = 10 \implies y = 2 \implies (0, 2)
\]
- The \(x\)-intercept of \(4x + 5y = 10\) occurs when \(y = 0\):
\[
4x + 5(0) = 10 \implies 4x = 10 \implies x = 2.5 \implies (2.5, 0)
\]
- Thus, both equations do not share the same intercepts.

4. **Option D: All equations share the point \((0, 2)\)**.

- We already found that \((0, 2)\) is a solution for \(2x + 7y = 14\). For the other equations, we can test them:
- \(4x + 5y = 10\) at \((0, 2)\) yields:
\[
4(0) + 5(2) = 10 \quad \text{(True)}
\]
- For \(4x + 14y = 28\):
\[
4(0) + 14(2) = 28 \quad \text{(True)}
\]
- For \(8x + 10y = 20\):
\[
8(0) + 10(2) = 20 \quad \text{(True)}
\]
- All equations do share the point \((0, 2)\).

Because **both Option B** provides a valid transformation between the two systems, it is a clear and correct reason why the first system is equivalent to the second. **Option D** is also correct, but it does not provide a transformation method like Option B does.

Therefore, the best answer for why the systems are equivalent is **B**: "Both equations, \(2x + 7y = 14\) and \(4x + 5y = 10\), can be multiplied by \(1/2\) to produce the equivalent system of equations."