Answer:
x = 2, y = -2, AB = 142
Step-by-step explanation:
The fact that X is the midpoint gives you two relations:
AX = XB
AX + XB = AB
Since you have two unknowns, this number of equations is sufficient to find their values. Substituting the given expressions in the above equations, you have ...
- 8(3x+5) -3(y-7) -22x = 5x +y +23 -4(x-12)
- 8(3x+5) -3(y-7) -22x + 5x +y +23 -4(x-12) = 2x -5y +128
Simplifying the first of these can make simplifying the second one easier.
24x +40 -3y +21 -22x = 5x +y +23 -4x +48
2x -3y +61 = x +y +71 . . . . . . we can use this simplification
x -4y = 10 . . . . . . . . . . . . . . . . subtract x+y+61
Now, we can simplify the second equation to ...
2x -3y +61 +x +y +71 = 2x -5y +128
3x -2y +132 = 2x -5y +128 . . . . . simplify the left side
x +3y = -4 . . . . . . . . . . . . . . . add -2x+5y-132
Then the two equations we need to solve are ...
Subtracting the second from the first, we get
-7y = 14
y = -2
Substituting into the first of these simplified equations, we get
x -4(-2) = 10 . . . . substitute for y
x +8 = 10 . . . . . . .evaluate
x = 2 . . . . . . . . . .subtract 8
So, the solution is (x, y) = (2, -2).
Now, the values of AX and XB are ...
AX = 2x -3y +61 = 2·2 -3(-2) +61 = 71
XB = x+y+71 = 2 +(-2) +71 = 71 . . . . . . . . matches AX, a good sign
AB = 2x -5y +128 = 2·2 -5(-2) +128 = 142 . . . . = AX+XB, another good sign
The desired values are x = 2, y = -2, AB = 142.
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You check the answer by filling the values into the expressions given in the problem statement and seeing if you get consistent results. Here, we used the simplified expressions, rather than the original expressions, so if we did the simplification wrong, we may have the wrong answer. It is always best to use the original equations. (A machine solver working with the original equations confirms our result, so "confidence is high.")
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