9 3x + 4y = -23 2y - x = -19 What is the solution (x, y) to the system of equations above? A) (-5,-2) B) (3,-8) C) (4,-6) D) (9,-6)

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Let's solve this system of linear equations. We have two equations: 3x plus 4y equals negative 23, and 2y minus x equals negative 19. We need to find the values of x and y that satisfy both equations simultaneously. Looking at the graph, we can see that the two lines intersect at the point (3, -8). Let's verify this algebraically. Let's solve this system algebraically using the substitution method. First, we'll solve the second equation for x. From 2y minus x equals negative 19, we get x equals 19 plus 2y. Next, we substitute this expression for x into the first equation. 3 times (19 plus 2y) plus 4y equals negative 23. Simplifying, we get 57 plus 10y equals negative 23. Solving for y, we get 10y equals negative 80, so y equals negative 8. Finally, we substitute y equals negative 8 back into our expression for x. x equals 19 plus 2 times negative 8, which gives us x equals 3. Therefore, the solution is the point (3, negative 8), which matches option B. Now let's solve the same system using the elimination method. First, we rearrange the second equation to get negative x plus 2y equals negative 19. Next, we multiply this equation by 3 to get negative 3x plus 6y equals negative 57. Now we can add this to the first equation, 3x plus 4y equals negative 23, to eliminate the x terms. Adding the equations gives us 10y equals negative 80, so y equals negative 8. Finally, we substitute y equals negative 8 back into the first equation. 3x plus 4 times negative 8 equals negative 23. Simplifying, we get 3x minus 32 equals negative 23, which gives us 3x equals 9, so x equals 3. Therefore, the solution is again (3, negative 8), confirming our answer is option B. Let's verify our solution by substituting the values x equals 3 and y equals negative 8 back into the original equations. For the first equation, 3x plus 4y equals negative 23, we get 3 times 3 plus 4 times negative 8, which is 9 minus 32, giving us negative 23. This matches the right side of the equation. For the second equation, 2y minus x equals negative 19, we get 2 times negative 8 minus 3, which is negative 16 minus 3, giving us negative 19. This also matches the right side. Since our solution satisfies both equations, we confirm that the answer is B, the point (3, negative 8). Let's summarize what we've learned about solving systems of linear equations. First, we can solve these systems using graphical, substitution, or elimination methods. The graphical method involves finding the intersection point of the lines representing each equation. The substitution method requires expressing one variable in terms of the other and then substituting this expression. The elimination method combines equations to eliminate one variable and solve for the other. Finally, it's always important to verify your solution by substituting the values back into the original equations. For our problem, we confirmed that the solution is the point (3, negative 8), which corresponds to option B.