To solve the given system of equations using the linear combination (or addition) method, let's analyze the steps involved:
We start with the system of equations:
1. [tex]\( 3x + 5y = 2 \)[/tex]
2. [tex]\( 6x + 10y = 8 \)[/tex]
Laura correctly multiplies the first equation by -2:
[tex]\[ -2 \times (3x + 5y) = -2 \times 2 \][/tex]
This simplifies to:
[tex]\[ -6x - 10y = -4 \][/tex]
Next, she adds the modified first equation to the second equation:
[tex]\[ (-6x - 10y) + (6x + 10y) = -4 + 8 \][/tex]
Combining the left-hand side terms:
[tex]\[ -6x - 10y + 6x + 10y = 0 \][/tex]
This simplifies to:
[tex]\[ 0 = -4 + 8 \][/tex]
And combining the right-hand sides:
[tex]\[ 0 = 4 \][/tex]
The resulting equation [tex]\( 0 = 4 \)[/tex] is a contradiction, as [tex]\( 0 \)[/tex] cannot equal [tex]\( 4 \)[/tex].
Since we arrived at a contradiction, this means that the system of equations is inconsistent and has no solutions.
Thus, the number of solutions to the system of equations is:
[tex]\[ 0 \][/tex]
