Depth · one concept from first principles · step 2 of 4
Two lines through the origin meeting twice coincide
The two lines and meet at some point other than , . Find the value of the constant .
Expressions
Your answer
How to solve
Strategy Organize Information in More Ways — Tidy both equations first. The constant terms vanish, so both lines go through the origin. Two distinct lines share at most one point, and the origin already uses it up; a second common point forces the lines to be the same line.
1STEP 1
Rearrange both equations
Move every term to the left side.
(a-3)x+2y=0, (a-1)x+3y=0
Rewriting reveals what the original forms hid: there is no constant term.
8.EE.C.8Organize Information In More Ways2STEP 2
Both lines pass through the origin
With no constant term, both lines pass through (0, 0). Two different lines cross at most once, so they must coincide.
Picture two lines through one point: unless they lie on top of each other, they spread apart and never meet again.
8.EE.C.8Draw A DiagramTwo lines through the origin that also meet somewhere else must be the same line.
Why?
With no constant term, , makes both and true, so both lines pass through the origin.
🧱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.
Why?
Writing the lines as and , a common point has , that is .
Why?
groups of minus groups of is groups of .
🧱Distributive propertya groups of (b and c) is a groups of b plus a groups of c — you can split the group.
Why?
A product is zero only if a factor is zero, so either , which is the origin, or , which makes the two lines the same.
🧱A product is zero only if a factor is zeroIf two numbers multiply to zero then at least one of them was zero, so a factored equation hands you one case per factor.
3STEP 3
Make the lines identical
Two lines through the origin coincide when their slopes are equal. From line 1, y=-((a-3)/2)x; from line 2, y=-((a-1)/3)x.
(a-3)/2=(a-1)/3, 3a-9=2a-2, a=7
Equal slopes plus a shared point means the same line.
8.EE.C.8Convert To AlgebraAnswer
a=7
With a=7 the lines are 4x+2y=0 and 6x+3y=0; both simplify to y=-2x, so they share every point, for example (1, -2).
Takeaway
Two lines can share one point or all of them -- never exactly two.
- Rearrange both equations
- Both lines pass through the origin
- Make the lines identical
Where next?
Level upsuggested
Another one like thissuggested
Step back and solidifysuggested
A parent dashboard for the family lives at sensimlab.com.