Depth · one concept from first principles · step 2 of 4
Find the common point from the known lines
The four lines , , , and all meet at one point. Find the values of the constants and .
Expressions
Your answer
How to solve
Strategy Change Focus / Count the Complement — Change focus from the unknown constants to the common point. The two lines with no constants already determine that point. Once it is known, plugging it into the other two lines gives a system whose unknowns are a and b.
1STEP 1
Find the point from the two known lines
Solve 3x+y=8 and x+3y=8. Adding them gives 4x+4y=16, so x+y=4; subtracting gives 2x-2y=0, so x=y.
x = y = 2
Two different lines meet in exactly one point, so this is the shared point of all four.
8.EE.C.8Change Focus Count The Complement2STEP 2
Put the point into the lines with constants
The point (2, 2) is on ax+by=4 and on 2ax-3by=-22.
2a + 2b = 4, 4a - 6b = -22
Now a and b are the unknowns of a new linear system.
8.EE.C.8Convert To AlgebraThe shared point makes both and true, which gives and .
Why?
The shared point lies on and on , and adding or subtracting these two balanced equations keeps them balanced, which forces and , so the point is .
🧱Do the same to both sides and equality holdsA balanced scale stays level when you take the same amount off both pans or add the same amount to both.
Why?
A point is on a line exactly when its and make that line's equation come out equal, so putting , into each line with constants must give a true statement.
🧱A solution makes the equation trueA solution is a number that, put in for the unknown, makes both sides come out equal; that is all being a solution means.
3STEP 3
Simplify both equations
Divide the first by 2 and the second by 2.
a + b = 2, 2a - 3b = -11
Smaller coefficients mean fewer arithmetic slips.
8.EE.C.8Convert To Algebra4STEP 4
Solve for a and b
Substitute a = 2 - b into the second equation.
2(2-b) - 3b = -11 → 4 - 5b = -11 → b = 3, a = 2 - 3 = -1
One substitution finishes the system.
8.EE.C.8Convert To AlgebraAnswer
a=-1, b=3
With a=-1, b=3 the lines are -x+3y=4 and -2x-9y=-22. At (2, 2): -2+6=4 and -4-18=-22. Both pass through (2, 2), as do 3x+y=8 and x+3y=8.
Takeaway
When several lines share a point, let the fully known lines find the point first.
- Find the point from the two known lines
- Put the point into the lines with constants
- Simplify both equations
- Solve for a and b
Where next?
Level upsuggested
Another one like thissuggested
Step back and solidifysuggested
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