Question

Consider the system of equations.

5x + 2y = 6,

10x + 4y = 12

Which equation is equivalent to the first equation of the system and can be used to solve the system using the linear combination method?



Which answer describes the number of solutions to the system of equations?

Answer 1

To analyze the given system of equations:

1. **First equation:**
\[
5x + 2y = 6
\]

2. **Second equation:**
\[
10x + 4y = 12
\]

We can first check if the second equation is a multiple of the first equation. If we multiply the first equation by 2, we get:

\[
2(5x + 2y) = 2(6) \\
10x + 4y = 12
\]

As we see, the second equation is indeed just twice the first equation. This means that both equations represent the same line in a two-dimensional space. Therefore, we can conclude that they are equivalent equations.

### Identifying an Equivalent Equation

To find an equation that is equivalent to the first one, we can directly use the original equation as stated, or simplify it. However, since you requested an equivalent form that allows the linear combination method, we can keep it as it is or express it in a different format. One possible rearrangement could be:

\[
2y = 6 - 5x \quad (isolate \, 2y \,from \, 5x + 2y = 6)
\]
or
\[
y = 3 - \frac{5}{2}x \quad (divide \,by\, 2 \,to \,solve\, for \,y)
\]

Each of these is also a valid equation that represents the same line.

### Number of Solutions

Since both equations represent the same line, they intersect at infinitely many points along that line. This means, regarding the number of solutions:

- The system has **infinitely many solutions** because the two equations are dependent (one is simply a multiple of the other).

### Summary:
- An equivalent equation to the first is \(2y = 6 - 5x\) or \(y = 3 - \frac{5}{2}x\).
- The number of solutions to the system is **infinitely many**.