If (Xo,Yo) is the unique solution to a system, how does shifting both lines vertically by 'c' units affect the solution? Prove algebraically
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Step-by-Step Solution
Step 1: Define the original system of equations
Let's start by defining a general system of two linear equations with two variables, x and y. We assume this system has a unique solution (Xo,Yo).
Step 2: Shift the lines vertically by 'c' units
Shifting a line vertically by 'c' units means replacing y with (y-c) in the equation. This is because if a point (x, y) was on the original line, then the point (x, y+c) will be on the shifted line. Substituting y′=y+c (or y=y′−c) into the original equation gives the new equation in terms of y′.
Step 3: Rearrange the shifted equations
Now, we expand the terms in the shifted equations and move the constant terms involving c to the right-hand side. This gives us a new system of equations.
Step 4: Compare with the original system
We can see that the coefficients of x and y (a1,b1,a2,b2) remain unchanged. Only the constant terms on the right-hand side have been modified. This means the slopes of the lines are preserved, and since the original system had a unique solution, the new system will also have a unique solution.
Step 5: Determine the new solution
Since (Xo,Yo) is the solution to the original system, it satisfies the original equations. If we substitute x=Xo and y=Yo+c into the new system, we find that these values satisfy the new equations. This means the x -coordinate of the solution remains the same, but the y -coordinate increases by c.