To determine which set of coordinates satisfies the system of equations:
[tex]\[ y = x - 3 \][/tex]
[tex]\[ y = -2x + 1 \][/tex]
we need to solve this system step-by-step:
1. Set the two equations equal to each other since they both equal [tex]\( y \)[/tex].
[tex]\[ x - 3 = -2x + 1 \][/tex]
2. Combine like terms to isolate [tex]\( x \)[/tex].
[tex]\[ x + 2x - 3 = 1 \][/tex]
[tex]\[ 3x - 3 = 1 \][/tex]
3. Add 3 to both sides of the equation to isolate the term with [tex]\( x \)[/tex].
[tex]\[ 3x = 4 \][/tex]
4. Divide both sides by 3 to solve for [tex]\( x \)[/tex].
[tex]\[ x = \frac{4}{3} \][/tex]
[tex]\[ x = 1.3333333333333333 \][/tex]
5. Substitute [tex]\( x = \frac{4}{3} \)[/tex] back into one of the original equations to solve for [tex]\( y \)[/tex]. Using the first equation:
[tex]\[ y = x - 3 \][/tex]
[tex]\[ y = \frac{4}{3} - 3 \][/tex]
[tex]\[ y = \frac{4}{3} - \frac{9}{3} \][/tex]
[tex]\[ y = \frac{4 - 9}{3} \][/tex]
[tex]\[ y = \frac{-5}{3} \][/tex]
[tex]\[ y = -1.6666666666666667 \][/tex]
Therefore, the coordinates that satisfy the system of equations are:
[tex]\[
\left( 1.3333333333333333, -1.6666666666666667 \right)
\][/tex]
This pair of coordinates satisfies the given system of equations. The answer is whichever option matches these coordinates, perhaps labeled as:
[tex]\[ B. \quad \left( 1.3333333333333333, -1.6666666666666667 \right) \][/tex]
Hence, the correct choice is [tex]\( B \)[/tex].
