Let's solve this system of linear equations. We have 3x plus 4y equals negative 23, and 2y minus x equals negative 19.
Step 1: We'll solve for x in terms of y using the second equation. From 2y minus x equals negative 19, we get x equals 19 plus 2y.
Step 2: Now we substitute this expression for x into the first equation. 3 times the quantity 19 plus 2y, plus 4y equals negative 23. Simplifying, we get 57 plus 6y plus 4y equals negative 23. This gives us 57 plus 10y equals negative 23. Solving for y, we get y equals negative 8.
Step 3: Now we substitute y equals negative 8 back into our expression for x. We get x equals 19 plus 2 times negative 8, which is 19 minus 16, giving us x equals 3.
Therefore, the solution to the system is x equals 3 and y equals negative 8. Looking at the given options, the answer is B.
Now, let's verify our solution by substituting x equals 3 and y equals negative 8 back into the original equations.
For the first equation, 3x plus 4y equals negative 23, we substitute x equals 3 and y equals negative 8. We get 3 times 3 plus 4 times negative 8, which is 9 plus negative 32, giving us negative 23. This matches the right side of the equation.
For the second equation, 2y minus x equals negative 19, we substitute y equals negative 8 and x equals 3. We get 2 times negative 8 minus 3, which is negative 16 minus 3, giving us negative 19. This also matches the right side of the equation.
Since both equations are satisfied by x equals 3 and y equals negative 8, we have verified that this is indeed the correct solution. Therefore, the answer is B.
Let's visualize the solution graphically. We have two equations: 3x plus 4y equals negative 23, shown in blue, and 2y minus x equals negative 19, shown in red.
Each equation represents a line in the xy-plane. The solution to the system is the point where these two lines intersect.
From the graph, we can see that the intersection point is at x equals 3 and y equals negative 8. This confirms our algebraic solution.
Therefore, the solution to the system of equations is the point 3, negative 8, which corresponds to option B.
Let's solve the same system using an alternative method: elimination. We'll start by rearranging the second equation to get x minus 2y equals 19.
Next, we add the first equation and the rearranged second equation. This gives us 3x plus 4y plus x minus 2y equals negative 23 plus 19, which simplifies to 4x plus 2y equals negative 4.
Now we solve for x in terms of y. From 4x plus 2y equals negative 4, we get x equals negative 1 minus y over 2.
We substitute this expression for x into the rearranged second equation. After simplifying, we get negative 5y over 2 equals 20, which gives us y equals negative 8.
Finally, we substitute y equals negative 8 back into our expression for x. We get x equals negative 1 minus negative 8 over 2, which simplifies to x equals 3.
So using the elimination method, we again find that the solution is x equals 3 and y equals negative 8, confirming our answer is option B.
Let's summarize what we've learned. We solved a system of two linear equations: 3x plus 4y equals negative 23, and 2y minus x equals negative 19.
We used multiple methods to solve this system: substitution, elimination, and graphical methods. All three approaches confirmed that the solution is x equals 3 and y equals negative 8.
We verified our solution by substituting x equals 3 and y equals negative 8 back into the original equations. Both equations were satisfied, confirming our answer.
Therefore, the solution to the system of equations is the point 3, negative 8, which corresponds to option B.