To determine how many solutions the system of equations has, we need to find if there is a point [tex]\((x, y)\)[/tex] that satisfies both equations simultaneously. Here are the equations given:
1. [tex]\(2y = 5x + 4\)[/tex]
2. [tex]\(y = 3x + 2\)[/tex]
To find the intersection point, we can substitute the expression for [tex]\(y\)[/tex] from the second equation into the first equation.
The second equation is:
[tex]\[ y = 3x + 2 \][/tex]
Substituting [tex]\( y = 3x + 2 \)[/tex] into the first equation gives us:
[tex]\[ 2(3x + 2) = 5x + 4 \][/tex]
Expanding and simplifying this:
[tex]\[ 6x + 4 = 5x + 4 \][/tex]
Next, we'll isolate [tex]\( x \)[/tex]:
[tex]\[ 6x + 4 - 5x = 4 \][/tex]
[tex]\[ x + 4 = 4 \][/tex]
By subtracting 4 from both sides of the equation:
[tex]\[ x = 0 \][/tex]
Now, with [tex]\( x = 0 \)[/tex], we substitute this back into the second equation to find [tex]\( y \)[/tex]:
[tex]\[ y = 3(0) + 2 \][/tex]
[tex]\[ y = 2 \][/tex]
So, the solution to the system of equations is [tex]\( (x, y) = (0, 2) \)[/tex].
Since we found one unique point that satisfies both equations, the system has one unique solution.
Thus, the number of solutions is:
B. One
